Comapbility testing is nothing but

checking the compabiltiy between browsers, operating 
systems.

To check whether our application supports customer expected 
platforms or not?

Checking the application on different platforms (Operating 
System) with different browsers,the main aim of 
compatibility testing is to test weather application can 
run successfully on different OS.

Three answers are correct.But Swaroop answer is exellent.

Test the application between defferent browsers with 
different versions,and different OS (like windows 
9x/xp/2000..,linux),different Dbs,and differnt networks

I agree with Swaroop Kumar.  
Compatibility should be on the following also:

-Testing On various client hardware configurations
-Using different memory sizes and hard drive space
-On various Operating Systems
-In different network environments
-With different printers and peripherals

Checking the application on different combination of most 
widely used Browsers, Operating systems, its different 
versions and also on differnt networks.