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