An API is an
Application Programming Interface. It's just the interface that a piece of software exposes so that other pieces of software can use it.
Probably you are not looking for the definition of what an API is, but for suggestions on how to design a good API for your software. There's ofcourse a lot that can be said about that subject.
One of the hardest things when designing an API is that it needs to be future-proof: if you develop a new version of your software in the future, you probably don't want to change the API in such a way that it's incompatible with older versions. Your clients (people who are using the API) won't be happy if they have to modify their programs to work with your new version.
In the standard Java API, some methods are marked deprecated (with the @deprecated Javadoc tag or the @Deprecated annotation) to indicate that the method should not be used anymore because the API has changed. Note that Sun until now has deprecated a number of classes, interfaces and methods in the standard Java API, but they've never actually removed any of those deprecated things. If they'd do that, they'd break compatibility with software that was written with the older version of the API.
So you have to be very careful to design an API for your software, and you'll need to think especially about how future-proof your API is going to be.
