When do you use the Bridge Pattern? How is it different from Adapter pattern?

There's a combination of Federico's and John's answers.

When:

                   ----Shape---
                  /            \
         Rectangle              Circle
        /         \            /      \
BlueRectangle  RedRectangle BlueCircle RedCircle

Refactor to:

          ----Shape---                        Color
         /            \                       /   \
Rectangle(Color)   Circle(Color)           Blue   Red

The Bridge pattern is an application of the old advice, "prefer composition over inheritance". It becomes handy when you must subclass different times in ways that are orthogonal with one another. Say you must implement a hierarchy of colored shapes. You wouldn't subclass Shape with Rectangle and Circle and then subclass Rectangle with RedRectangle, BlueRectangle and GreenRectangle and the same for Circle, would you? You would prefer to say that each Shape has a Color and to implement a hierarchy of colors, and that is the Bridge Pattern. Well, I wouldn't implement a "hierarchy of colors", but you get the idea...

When:

        A
     /     \
    Aa      Ab
   / \     /  \
 Aa1 Aa2  Ab1 Ab2

Refactor to:

     A         N
  /     \     / \
Aa(N) Ab(N)  1   2

A classic example of the Bridge pattern is used in the definition of shapes in an UI environment (see the Bridge pattern Wikipedia entry). The Bridge pattern is a composite of the Template and Strategy patterns.

It is a common view some aspects of the Adapter pattern in the Bridge pattern. However, to quote from this article:

At first sight, the Bridge pattern looks a lot like the Adapter pattern in that a class is used to convert one kind of interface to another. However, the intent of the Adapter pattern is to make one or more classes' interfaces look the same as that of a particular class. The Bridge pattern is designed to separate a class's interface from its implementation so you can vary or replace the implementation without changing the client code.

In my experience, Bridge is a quite often recurring pattern, because it's the solution whenever there are two orthogonal dimensions in the domain. E.g. shapes and drawing methods, behaviours and platforms, file formats and serializers and so forth.

And an advice: always think of design patterns from the conceptual perspective, not from the implementation perspective. From the right point of view, Bridge cannot be confused with Adapter, because they solve a different problem, and composition is superior to inheritance not because of the sake of itself, but because it allows to handle orthogonal concerns separately.