Purpose: Adapters are used to enable objects with different interfaces to communicate with each other which convert the interface of a class into another interface clients expect thus let classes work together without imcompatible interfaces problem.

Object Adapter:

-- use compositional technique
-- by composing interface B's instance within interface A (target interface, adapter) and implement interface A in terms of B's interface(adaptee). Interface A is Object Adapter which enables the client (the class or object to use Interface A) and the adaptee(Interface B) to be completely decoupled from each other.

Class Adapter:

-- use multiple inheritance
-- Notice that Java is not supporting true multiple inheritance, there must be one inheritance is from Java Interface(implementation) not Java class. So take above Object Adapter as example, if client want to access Interface B, the adpater Interface A will inherit from class(Class Adapter) which Client can access, and inherit the implementation of Interface B(Adaptee). So Interface A can have a funtion to call functions in Interface B, and so that Interface will not have an object of Interface B, that is the difference from Object Adapter method.

reference:
http://www.exciton.cs.rice.edu/JavaResources/DesignPatterns/adapter.htm