If there's one reason to continue using Twitter outside the desktop environment other than your all-consuming addition to checking updates, it's Tweetbot. The little robot bird that sits on your iPhone's display begging you to tap, showing you that using Twitter can actually be more fun on the iPhone than it is anywhere else. This app uses a selection of gestures to allow you to interact with each Tweet, has a perfectly excellent aesthetic to the entirety of it, and has customizable tabs to make your experience just what you want it to be.
Once you sign in, you'll see your Twitter feed on the leftmost tab, your mentions next, messages, then two custom tabs. In these tabs you'll have a selection of Favorites, Lists, Retweets, Mute Filters, Search, and your Profile. These two buttons have all of these options but will not repeat the same option twice. It's kind of magical like that. Your interface gives you all the options you get on the desktop web browser version of Twitter and more - connections!
When you double-tap on a users profile, you'll be taken to their profile on Favstar. If you double-tap on the direct messages tab, you get the option of marking all of your DMs as read at once. There's even a triple-tap for each tweet, your options being several including replying or starring that tweet as a favorite. To access a Tweet's entire conversation, you've only to swipe to the right. If you want to see any replies that a single tweet has gotten, swipe left.
This entire app also has a lovely set of sound effects for each action, each of them a simple foosh, tic, or bamp, making the whole set of interactions fun to initiate. This application is currently on the market for both iPhone and iPad and you can pick up the iPhone version for a cool $2.99 - go grab it!