If you are writing tests to check new code in tandem with writing that code (either via TDD, or some other code-test loop), or are writing tests for existing code, you can (and usually will) find and fix bugs. Likewise if you are investigating a problem and write one or more test cases to check the behaviour.

Once those tests have been written, then they act as regression tests like the parent comment notes.

On "unit tests", I view the behaviour of the class/function and all its dependencies as a single unit for the purpose of testing. I've never liked the idea of needing to mock out a class in order to test another class, just because the class being tested makes use of it. The only case where mocks/stubs/etc. should be used is when interacting with an external component like a database or HTTP API. -- You don't see people that do this mocking out a list, string or other library classes, so why should project classes be any different.