On a PC system, one normally has a means of generating input and checking the output via a test harness program. However, on an embedded system this may involve hardware. There's no magic to all of this - testing tends to be hard work. It all comes down to generating inputs (be it software or hardware) and checking the results (via hardware or software). Embedded systems is a broad term, so the methods used for testing is broad also. Testing an automotive ECU would be totally different to testing an internet appliance. The concepts are the same, the actual method is quite different.

For example, one technique would be to simulate the complete hardware on the PC and write test code that would apply the stimulus. This may allow more extensive diagnostic information to be available.

Regression testing is one method of ensuring software quality, there are other methods that can assist this task. Version control I would consider mandatory, code reviews can also be most helpful.

If you have a specific case, then present it then we can give a more specific answer.