The best definition I've found so far is one by James Shore:

"Dependency Injection" is a 25-dollar term for a 5-cent concept. [...] Dependency injection means giving an object its instance variables. [...].

There is an article by Martin Fowler that may prove useful, too.

Dependency injection is basically providing the objects that an object needs (its dependencies) instead of having it construct them itself. It's a very useful technique for testing, since it allows dependencies to be mocked or stubbed out.

Dependencies can be injected into objects by many means (such as constructor injection or setter injection). One can even use specialized dependency injection frameworks (e.g. Spring) to do that, but they certainly aren't required. You don't need those frameworks to have dependency injection. Instantiating and passing objects (dependencies) explicitly is just as good an injection as injection by framework.

Short Answer

Use $this to refer to the current object. Use self to refer to the current class. In other words, use $this->member for non-static members, use self::$member for static members.

Full Answer

Here is an example of correct usage of $this and self for non-static and static member variables:

<?php
class X {
    private $non_static_member = 1;
    private static $static_member = 2;

    function __construct() {
        echo $this->non_static_member . ' '
           . self::$static_member;
    }
}

new X();
?>

Here is an example of incorrect usage of $this and self for non-static and static member variables:

<?php
class X {
    private $non_static_member = 1;
    private static $static_member = 2;

    function __construct() {
        echo self::$non_static_member . ' '
           . $this->static_member;
    }
}

new X();
?>

Here is an example of polymorphism with $this for member functions:

<?php
class X {
    function foo() {
        echo 'X::foo()';
    }

    function bar() {
        $this->foo();
    }
}

class Y extends X {
    function foo() {
        echo 'Y::foo()';
    }
}

$x = new Y();
$x->bar();
?>

Here is an example of suppressing polymorphic behaviour by using self for member functions:

<?php
class X {
    function foo() {
        echo 'X::foo()';
    }

    function bar() {
        self::foo();
    }
}

class Y extends X {
    function foo() {
        echo 'Y::foo()';
    }
}

$x = new Y();
$x->bar();
?>

The idea is that $this->foo() calls the foo() member function of whatever is the exact type of the current object. If the object is of type X, it thus calls X::foo(). If the object is of type Y, it calls Y::foo(). But with self::foo(), X::foo() is always called.

From http://www.phpbuilder.com/board/showthread.php?t=10354489:

By http://board.phpbuilder.com/member.php?145249-laserlight