OAuth can be a tricky concept to wrap your head around at first, but with the Twitter API now requiring its use, it is something you need to understand before creating a Twitter application. This tutorial: Creating a Twitter OAuth Application will introduce you to OAuth, and walk you through the process of creating a basic application.

Requirements: OAuth
Demo: http://net.tutsplus.com/tutorials/php/creating-a-twitter-oauth-application/
License: License Free