A very non technical explaination lacking any details.... Here comes..
Blackbox Testing : Testing an application without any knowledge of how the internal application works
Whitebox Testing: Testing an application with knowledge of how the internal works, such as by having the source code side by side while you are doing your test.
Unit Testing: This is where you create tests which interact directly with your application. You would check a function in your application and assert that the response should return with value X. Unit Tests are usually, but not always created by the developers themselves as well, whereas if a company does whitebox and blackbox testing, it can be done by anyone.
This is a very basic explaination.
