First you will want a way to use the Twitter API. Go get MGTwitterEngine for that (I won't talk about that here since lots of people have explained it well). Get that and install it and play with it. You should be posting tweets very quickly from your app with Basic Auth.
Ok, once you get tweets sending and can retrieve tweets with Matt's code (MGTwitterEngine) you might want to start posting your tweets with your own app name and website. To do this you will need to use OAuth. See the Twitter API FAQ about OAuth and Twitter.
The best way to get this working on the iPhone is to use Ben Gottlieb's Twitter-OAuth-iPhone. This is a awesome piece of work by Ben, piecing together several examples, frameworks, and the OAuth process to make using OAuth from the iPhone a piece of cake.
Follow Ben's readme and sample project and you will be up and running with application branded tweets in no time.