Here are some resources that will help you understand the basics of all our APIs. If you haven't already, make sure that you have familiarized yourself with the Developer Rules of the Road. Check out the OAuth section to learn more about how we do authentication and authorization.
There are four main "objects" that you'll encounter in the API: Tweets, Users, Entities (see also Entities in Objects), and Places. See the anatomy of these objects, and learn about properties like Twitter IDs or Place Attributes to know what to expect.