There are two broad types of Twitter APIs: authenticated and non-authenticated. The type is mentioned in the API docs.
If your application is lucky enough to need only non-auth APIs then all you need to do is make HTTP requests and parse the result, in your favourite language.
Authenticated APIs need more work (especially with the move to OAuth). As others have replied, you should look at existing libraries as a starting point. They are all linked in the wiki. Dispatch wasn't listed,
http://databinder.net/dispatch/About which is great for working with Scala.
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Harshad RJ
http://hrj.wikidot.com