Unlike inheritance; if a trait has static properties, each class using that trait has independent instances of those properties.

Example using parent class:
<?php
class TestClass {
    public static $_bar;
}
class Foo1 extends TestClass { }
class Foo2 extends TestClass { }
Foo1::$_bar = 'Hello';
Foo2::$_bar = 'World';
echo Foo1::$_bar . ' ' . Foo2::$_bar; // Prints: World World
?>

Example using trait:
<?php
trait TestTrait {
    public static $_bar;
}
class Foo1 {
    use TestTrait;
}
class Foo2 {
    use TestTrait;
}
Foo1::$_bar = 'Hello';
Foo2::$_bar = 'World';
echo Foo1::$_bar . ' ' . Foo2::$_bar; // Prints: Hello World
?>

The best way to understand what traits are and how to use them is to look at them for what they essentially are:  language assisted copy and paste.

If you can copy and paste the code from one class to another (and we've all done this, even though we try not to because its code duplication) then you have a candidate for a trait.

Note that the "use" operator for traits (inside a class) and the "use" operator for namespaces (outside the class) resolve names differently. "use" for namespaces always sees its arguments as absolute (starting at the global namespace):

<?php
namespace Foo\Bar;
use Foo\Test;  // means \Foo\Test - the initial \ is optional
?>

On the other hand, "use" for traits respects the current namespace:

<?php
namespace Foo\Bar;
class SomeClass {
    use Foo\Test;   // means \Foo\Bar\Foo\Test
}
?>

Together with "use" for closures, there are now three different "use" operators. They all mean different things and behave differently.

Another difference with traits vs inheritance is that methods defined in traits can access methods and properties of the class they're used in, including private ones.

For example:
<?php
trait MyTrait
{
  protected function accessVar()
  {
    return $this->var;
  }

}

class TraitUser
{
  use MyTrait;

  private $var = 'var';

  public function getVar()
  {
    return $this->accessVar();
  }
}

$t = new TraitUser();
echo $t->getVar(); // -> 'var'                                                                                                                                                                                                                          

?>

It may be worth noting here that the magic constant __CLASS__ becomes even more magical - __CLASS__ will return the name of the class in which the trait is being used.

for example

<?php
trait sayWhere {
    public function whereAmI() {
        echo __CLASS__;
    }
}

class Hello {
    use sayWHere;
}

class World {
    use sayWHere;
}

$a = new Hello;
$a->whereAmI(); //Hello

$b = new World;
$b->whereAmI(); //World
?>

The magic constant __TRAIT__ will giev you the name of the trait

Keep in mind; "final" keyword is useless in traits when directly using them, unlike extending classes / abstract classes.

<?php
trait Foo {
    final public function hello($s) { print "$s, hello!"; }
}
class Bar {
    use Foo;
    // Overwrite, no error
    final public function hello($s) { print "hello, $s!"; }
}

abstract class Foo {
    final public function hello($s) { print "$s, hello!"; }
}
class Bar extends Foo {
    // Fatal error: Cannot override final method Foo::hello() in ..
    final public function hello($s) { print "hello, $s!"; }
}
?>

But this way will finalize trait methods as expected;

<?php
trait FooTrait {
    final public function hello($s) { print "$s, hello!"; }
}
abstract class Foo {
    use FooTrait;
}
class Bar extends Foo {
    // Fatal error: Cannot override final method Foo::hello() in ..
    final public function hello($s) { print "hello, $s!"; }
}
?>

Here is an example how to work with visiblity and conflicts.

<?php

trait A
{
    private function smallTalk()
    {
        echo 'a';
    }

    private function bigTalk()
    {
        echo 'A';
    }
}

trait B
{
    private function smallTalk()
    {
        echo 'b';
    }

    private function bigTalk()
    {
        echo 'B';
    }
}

trait C
{
    public function smallTalk()
    {
        echo 'c';
    }

    public function bigTalk()
    {
        echo 'C';
    }
}

class Talker
{
    use A, B, C {
        //visibility for methods that will be involved in conflict resolution
        B::smallTalk as public;
        A::bigTalk as public;

        //conflict resolution
        B::smallTalk insteadof A, C;
        A::bigTalk insteadof B, C;

        //aliases with visibility change
        B::bigTalk as public Btalk;
        A::smallTalk as public asmalltalk;
        
        //aliases only, methods already defined as public
        C::bigTalk as Ctalk;
        C::smallTalk as cmallstalk;
    }

}

(new Talker)->bigTalk();//A
(new Talker)->Btalk();//B
(new Talker)->Ctalk();//C

(new Talker)->asmalltalk();//a
(new Talker)->smallTalk();//b
(new Talker)->cmallstalk();//c

A number of the notes make incorrect assertions about trait behaviour because they do not extend the class.

So, while "Unlike inheritance; if a trait has static properties, each class using that trait has independent instances of those properties.

Example using parent class:
<?php
class TestClass {
    public static $_bar;
}
class Foo1 extends TestClass { }
class Foo2 extends TestClass { }
Foo1::$_bar = 'Hello';
Foo2::$_bar = 'World';
echo Foo1::$_bar . ' ' . Foo2::$_bar; // Prints: World World
?>

Example using trait:
<?php
trait TestTrait {
    public static $_bar;
}
class Foo1 {
    use TestTrait;
}
class Foo2 {
    use TestTrait;
}
Foo1::$_bar = 'Hello';
Foo2::$_bar = 'World';
echo Foo1::$_bar . ' ' . Foo2::$_bar; // Prints: Hello World
?>"

shows a correct example, simply adding
<?php
require_once('above');
class Foo3 extends Foo2 {
}
Foo3::$_bar = 'news';
echo Foo1::$_bar . ' ' . Foo2::$_bar . ' ' . Foo3::$_bar; 

// Prints: Hello news news

I think the best conceptual model of an incorporated trait is an advanced insertion of text, or as someone put it "language assisted copy and paste." If Foo1 and Foo2 were defined with $_bar, you would not expect them to share the instance. Similarly, you would expect Foo3 to share with Foo2, and it does.

Viewing this way explains away a lot of  the 'quirks' that are observed above with final, or subsequently declared private vars,

I have not seen this specific use case:

"Wanting to preserve action of parent class method, the trait one calling ::parent & also the child class mehod action".

// Child class.
use SuperTrait {
  initialize as initializeOr;
}
public function initialize(array &$element) {
  ...
  $this->initializeOr($element);
}
// Trait.
public function initialize(array &$element) {
  ...
  parent::initialize($element);
}
// Parent class.
public function initialize(array &$element) {
  ...
}

Note that you can omit a method's inclusion by excluding it from one trait in favor of the other and doing the exact same thing in the reverse way.

<?php

trait A {
    public function sayHello()
    {
        echo 'Hello from A';
    }

    public function sayWorld()
    {
        echo 'World from A';
    }
}

trait B {
    public function sayHello()
    {
        echo 'Hello from B';
    }

    public function sayWorld()
    {
        echo 'World from B';
    }
}

class Talker {
    use A, B {
        A::sayHello insteadof B;
        A::sayWorld insteadof B;
        B::sayWorld insteadof A;
    }
}

$talker = new Talker();
$talker->sayHello();
$talker->sayWorld();

?>

The method sayHello is imported, but the method sayWorld is simply excluded.

Adding to "atorich at gmail dot com":

The behavior of the magic constant __CLASS__ when used in traits is as expected if you understand traits and late static binding (http://php.net/manual/en/language.oop5.late-static-bindings.php).

<?php

$format = 'Class: %-13s | get_class(): %-13s | get_called_class(): %-13s%s';

trait TestTrait {
    public function testMethod() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
    
    public static function testStatic() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
}

trait DuplicateTrait {
    public function duplMethod() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
    
    public static function duplStatic() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
}

abstract class AbstractClass {
    
    use DuplicateTrait;
    
    public function absMethod() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
    
    public static function absStatic() {
        global $format;
        printf($format, __CLASS__, get_class(), get_called_class(), PHP_EOL);
    }
}

class BaseClass extends AbstractClass {
    use TestTrait;
}

class TestClass extends BaseClass { }

$t = new TestClass();

$t->testMethod();
TestClass::testStatic();

$t->absMethod();
TestClass::absStatic();

$t->duplMethod();
TestClass::duplStatic();

?>

Will output:

Class: BaseClass     | get_class(): BaseClass     | get_called_class(): TestClass    
Class: BaseClass     | get_class(): BaseClass     | get_called_class(): TestClass    
Class: AbstractClass | get_class(): AbstractClass | get_called_class(): TestClass    
Class: AbstractClass | get_class(): AbstractClass | get_called_class(): TestClass    
Class: AbstractClass | get_class(): AbstractClass | get_called_class(): TestClass    
Class: AbstractClass | get_class(): AbstractClass | get_called_class(): TestClass

Since Traits are considered literal "copying/pasting" of code, it's clear how the methods defined in DuplicateTrait give the same results as the methods defined in AbstractClass.

The difference between Traits and multiple inheritance is in the inheritance part.   A trait is not inherited from, but rather included or mixed-in, thus becoming part of "this class".   Traits also provide a more controlled means of resolving conflicts that inevitably arise when using multiple inheritance in the few languages that support them (C++).  Most modern languages are going the approach of a "traits" or "mixin" style system as opposed to multiple-inheritance, largely due to the ability to control ambiguities if a method is declared in multiple "mixed-in" classes.

Also, one can not "inherit" static member functions in multiple-inheritance.

About the (Safak Ozpinar / safakozpinar at gmail)'s great note, you can still have the same behavior than inheritance using trait with this approach : 
<?php

trait TestTrait {
    public static $_bar;
}

class FooBar {
    use TestTrait;
}

class Foo1 extends FooBar {

}
class Foo2 extends FooBar {

}
Foo1::$_bar = 'Hello';
Foo2::$_bar = 'World';
echo Foo1::$_bar . ' ' . Foo2::$_bar; // Prints: World World

As already noted, static properties and methods in trait could be accessed directly using trait. Since trait is language assisted c/p, you should be aware that static property from trait will be initialized to the value trait property had in the time of class declaration. 

Example:

<?php

trait Beer {
    protected static $type = 'Light';
    public static function printed(){
        echo static::$type.PHP_EOL;
    }
    public static function setType($type){
        static::$type = $type;
    }
}

class Ale {
    use Beer;
}

Beer::setType("Dark");

class Lager {
    use Beer;
}

Beer::setType("Amber");

header("Content-type: text/plain");

Beer::printed();  // Prints: Amber
Ale::printed();   // Prints: Light
Lager::printed(); // Prints: Dark

?>

(It's already been said, but for the sake of searching on the word "relative"...)

The "use" keyword to import a trait into a class will resolve relative to the current namespace and therefore should include a leading slash to represent a full path, whereas "use" at the namespace level is always absolute.

Simple singleton trait.

<?php

trait singleton {    
    /**
     * private construct, generally defined by using class
     */
    //private function __construct() {}
    
    public static function getInstance() {
        static $_instance = NULL;
        $class = __CLASS__;
        return $_instance ?: $_instance = new $class;
    }
    
    public function __clone() {
        trigger_error('Cloning '.__CLASS__.' is not allowed.',E_USER_ERROR);
    }
    
    public function __wakeup() {
        trigger_error('Unserializing '.__CLASS__.' is not allowed.',E_USER_ERROR);
    }
}

/**
 * Example Usage
 */

class foo {
    use singleton;
    
    private function __construct() {
        $this->name = 'foo';
    }
}

class bar {
    use singleton;
    
    private function __construct() {
        $this->name = 'bar';
    }
}

$foo = foo::getInstance();
echo $foo->name;

$bar = bar::getInstance();
echo $bar->name;

https://3v4l.org/mFuQE

1. no deprecate if same-class-named method get from trait
2. replace same-named method ba to aa in C

trait ATrait {
    public function a(){
        return 'Aa';
    }
}

trait BTrait {
    public function a(){
        return 'Ba';
    }
}

class C {
    use ATrait{
        a as aa;
    }
    use BTrait{
        a as ba;
    }
    
    public function a() {
        return static::aa() . static::ba();
    }
}

$o = new C;
echo $o->a(), "\n";

class D {
    use ATrait{
        ATrait::a as aa;
    }
    use BTrait{
        BTrait::a as ba;
    }
    
    public function a() {
        return static::aa() . static::ba();
    }
}

$o = new D;
echo $o->a(), "\n";

class E {
    use ATrait{
        ATrait::a as aa;
        ATrait::a insteadof BTrait;
    }
    use BTrait{
        BTrait::a as ba;
    }
    
    public function e() {
        return static::aa() . static::ba();
    }
}

$o = new E;
echo $o->e(), "\n";

class F {
    use ATrait{
        a as aa;
    }
    use BTrait{
        a as ba;
    }
    
    public function f() {
        return static::aa() . static::ba();
    }
}

$o = new F;
echo $o->f(), "\n";

AaAa 
AaBa 

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; E has a deprecated constructor in /in/mFuQE on line 48 
AaBa 

Fatal error: Trait method a has not been applied, because there are collisions with other trait methods on F in /in/mFuQE on line 65

If you want to resolve name conflicts and also change the visibility of a trait method, you'll need to declare both in the same line:

trait testTrait{
    
    public function test(){
        echo 'trait test';
    }
    
}

class myClass{
    
    use testTrait {
        testTrait::test as private testTraitF;
    }
    
    public function test(){
        echo 'class test';
        echo '<br/>';
        $this->testTraitF();
    }
    
}

$obj = new myClass(); 
$obj->test(); //prints both 'trait test' and 'class test'
$obj->testTraitF(); //The method is not accessible (Fatal error: Call to private method myClass::testTraitF() )

Traits are useful for strategies, when you want the same data to be handled (filtered, sorted, etc) differently.

For example, you have a list of products that you want to filter out based on some criteria (brands, specs, whatever), or sorted by different means (price, label, whatever). You can create a sorting trait that contains different functions for different sorting types (numeric, string, date, etc). You can then use this trait not only in your product class (as given in the example), but also in other classes that need similar strategies (to apply a numeric sort to some data, etc).

<?php
trait SortStrategy {
    private $sort_field = null;
    private function string_asc($item1, $item2) {
        return strnatcmp($item1[$this->sort_field], $item2[$this->sort_field]);
    }
    private function string_desc($item1, $item2) {
        return strnatcmp($item2[$this->sort_field], $item1[$this->sort_field]);
    }
    private function num_asc($item1, $item2) {
        if ($item1[$this->sort_field] == $item2[$this->sort_field]) return 0;
        return ($item1[$this->sort_field] < $item2[$this->sort_field] ? -1 : 1 );
    }
    private function num_desc($item1, $item2) {
        if ($item1[$this->sort_field] == $item2[$this->sort_field]) return 0;
        return ($item1[$this->sort_field] > $item2[$this->sort_field] ? -1 : 1 );
    }
    private function date_asc($item1, $item2) {
        $date1 = intval(str_replace('-', '', $item1[$this->sort_field]));
        $date2 = intval(str_replace('-', '', $item2[$this->sort_field]));
        if ($date1 == $date2) return 0;
        return ($date1 < $date2 ? -1 : 1 );
    }
    private function date_desc($item1, $item2) {
        $date1 = intval(str_replace('-', '', $item1[$this->sort_field]));
        $date2 = intval(str_replace('-', '', $item2[$this->sort_field]));
        if ($date1 == $date2) return 0;
        return ($date1 > $date2 ? -1 : 1 );
    }
}

class Product {
    public $data = array();
    
    use SortStrategy;
    
    public function get() {
        // do something to get the data, for this ex. I just included an array
        $this->data = array(
            101222 => array('label' => 'Awesome product', 'price' => 10.50, 'date_added' => '2012-02-01'),
            101232 => array('label' => 'Not so awesome product', 'price' => 5.20, 'date_added' => '2012-03-20'),
            101241 => array('label' => 'Pretty neat product', 'price' => 9.65, 'date_added' => '2012-04-15'),
            101256 => array('label' => 'Freakishly cool product', 'price' => 12.55, 'date_added' => '2012-01-11'),
            101219 => array('label' => 'Meh product', 'price' => 3.69, 'date_added' => '2012-06-11'),
        );
    }
    
    public function sort_by($by = 'price', $type = 'asc') {
        if (!preg_match('/^(asc|desc)$/', $type)) $type = 'asc';
        switch ($by) {
            case 'name':
                $this->sort_field = 'label';
                uasort($this->data, array('Product', 'string_'.$type));
            break;
            case 'date':
                $this->sort_field = 'date_added';
                uasort($this->data, array('Product', 'date_'.$type));
            break;
            default:
                $this->sort_field = 'price';
                uasort($this->data, array('Product', 'num_'.$type));
        }
    }
}

$product = new Product();
$product->get();
$product->sort_by('name');
echo '<pre>'.print_r($product->data, true).'</pre>';
?>

don't forget you can create complex (embedded) traits as well

<?php
trait Name {
  // ...
}
trait Address {
  // ...
}
trait Telephone {
  // ...
}
trait Contact {
  use Name, Address, Telephone;
}
class Customer {
  use Contact;
}
class Invoce {
  use Contact;
}
?>

I think it's obvious to notice that using 'use' followed by the traits name must be seen as just copying/pasting lines of code into the place where they are used.

Trait can not have the same name as class because it will  show: Fatal error: Cannot redeclare class

If you override a method which was defined by a trait, calling the parent method will also call the trait's override. Therefore if you need to derive from a class which has a trait, you can extend the class without losing the trait's functionality:

<?php

trait ExampleTrait
{
    public function output()
    {
        parent::output();
        echo "bar<br>";
    }
}

class Foo
{
    public function output()
    {
        echo "foo<br>";
    }
}

class FooBar extends Foo
{
    use ExampleTrait;
}

class FooBarBaz extends FooBar
{
    use ExampleTrait;
    public function output()
    {
        parent::output();
        echo "baz";
    }
}

(new FooBarBaz())->output();
?>

Output:
foo
bar
baz

/*
DocBlocks pertaining to the class or trait will NOT be carried over when applying the trait.

Results trying a couple variations on classes with and without DocBlocks that use a trait with a DocBlock
*/

<?php

/**
 * @Entity
 */
trait Foo
{
    protected $foo;
}

/**
 * @HasLifecycleCallbacks
 */
class Bar
{
    use \Foo;
    
    protected $bar;
}

class MoreBar
{
    use \Foo;
    
    protected $moreBar;
}

$w = new \ReflectionClass('\Bar');
echo $w->getName() . ":\r\n";
echo $w->getDocComment() . "\r\n\r\n";

$x = new \ReflectionClass('\MoreBar');
echo $x->getName() . ":\r\n";
echo $x->getDocComment() . "\r\n\r\n";

$barObj = new \Bar();
$y = new \ReflectionClass($barObj);
echo $y->getName() . ":\r\n";
echo $y->getDocComment() . "\r\n\r\n";

foreach($y->getTraits() as $traitObj) {
    echo $y->getName() . " ";
    echo $traitObj->getName() . ":\r\n";
    echo $traitObj->getDocComment() . "\r\n";
}

$moreBarObj = new \MoreBar();
$z = new \ReflectionClass($moreBarObj);
echo $z->getName() . " ";
echo $z->getDocComment() . "\r\n\r\n";

foreach($z->getTraits() as $traitObj) {
    echo $z->getName() . " ";
    echo $traitObj->getName() . ":\r\n";
    echo $traitObj->getDocComment() . "\r\n";
}

A note to 'Beispiel #9 Statische Variablen'. A trait can also have a static property:

trait Counter {
    static $trvar=1;

    public static function stfunc() {
        echo "Hello world!"
    }
}

class C1 {
    use Counter;
}

print "\nTRVAR: " . C1::$trvar . "\n";   //prints 1

$obj = new C1();
C1::stfunc();   //prints  Hello world!
$obj->stfunc();   //prints Hello world!

A static property (trvar) can only be accessed using the classname (C1).
But a static function (stfunc) can be accessed using the classname or the instance ($obj).