I've been trying to learn how to use the APIs available out there, but I can't seem to find any good book or tutorial out there. So... where can I start learning about using the available APIs (e.g. Twitter API)? I've only gone to Wikipedia so far.
You will never find a single source for all the APIs. There are far too many.
However, most API's have documentation, so just search for "Twitter API Documentation" or whatever API you want. Some have wiki's, some have introductions, others just have reference material. Nevertheless, its a good source for figuring out what methods and such you need to call.
AFAIK there are not books for this. Start taking a look at the api providers homepage. Mostly there is a page for developers or about the API.
For twitter this is http://dev.twitter.com/
When learning about APIs, I usually start off by reading code from open source projects, while trying to read the (often sparsely documented) documentation. There is usually no end-all tutorial for the API you're trying to use, so you need to get a general feel for the API, and take it step by step.
API is a general concept and every programming interface is referred as API. If you mean Win32 APIs try old versions of MSDN
Simply jump in and do it. The "Getting started" or "faq" page of an API is usually a good starting point.
If the code behind your API is open source, a good idea is to look if there are automated tests around. They might show you how to actually use the API, which might be easier than just reading the documentation.