Component testing is that in which we test those test objects which are separately testable and without integrating with other components. Component testing is done by the tester and always before component testing unit testing is conducted by the developer. Component testing is consider as module testing, because we are testing each module differently and effectively.
Compatibility testing is a type of non-functional testing. In this testing we test that our software is compatible with the computing environment. Such as operating system, Hardware platform, database, browser compatibility, other system software.