False negative / false positive tests

Software testing is not only running test cases but more of using a testers mindset and skills to find bugs and verify that the software is working correctly. However, usage of test cases (manual or automated) is essential to speed-up your testing and having reproducible tests for your regression testing. Obviously you want to be able to trust the tests - especially the outcome of these tests. However, regardless whether it is manual or automated testing, there are two terms you should understand and bear in mind. So what is it?