Expect everything to change...

Unit test: Specify and test one point of the contract of single method of a class. This should have a very narrow and well defined scope. Complex dependencies and interactions to the outside world are stubbed or mocked.

Integration test: Test the correct inter-operation of multiple subsystems. There is whole spectrum there, from testing integration between two classes, to testing integration with the production environment.

Smoke test (aka Sanity check): A simple integration test where we just check that when the system under test is invoked it returns normally and does not blow up. It is an analogy with electronics, where the first test occurs when powering up a circuit: if it smokes, it's bad.

Regression test: A test that was written when a bug was fixed. It ensures that this specific bug will not occur again. The full name is "non-regression test". It can also be a test made prior to changing an application to make sure the application provides the same outcome.

Acceptance test: Test that a feature or use case is correctly implemented. It is similar to an integration test, but with a focus on the use case to provide rather than on the components involved.

System test: Tests a system as a black box. Dependencies on other systems are often mocked or stubbed during the test (otherwise it would be more of an integration test).

Pre-flight check: Tests that are repeated in a production-like environment, to alleviate the 'builds on my machine' syndrome. Often this is realized by doing an acceptance or smoke test in a production like environment

Black-box testing: testing only the public interface with no knowledge of how the thing works.

Glass-box testing: testing all parts of a thing with full knowledge of how it works.