Java: Which library should I know?

You should definitely know the Google Collections Library. A very useful collection of classes which can be used in all sorts of places in an application. The library is licensed under the Apache License 2.0 so that shouldn’t be a problem, I think.

Personally I love the classes Preconditions, Lists, Sets, and Maps. Have a look at the other classes – there are some more. It’s absolutely worth it!