UPDATE: Tweetie is now the Official Twitter App for iPhone, simply called Twitter
The app so nice he made it Twice! Tweetie 2 is an award-winning Twitter app for the iPhone and iPod Touch, stowing away an unbelievable number of features into a simple, easy, attractive interface. Seriously!! Every pixel of this interface is precisely designed and intuitive to use.
Readtwit is a web service that finds links in your Twitter stream and formats them as an RSS feed. You may be saying "hold on here, I can get an RSS feed of my tweets from Twitter.com" but Readtwit has some goodies that make it a worthwhile service.
First of all, it only loads tweets with links in them. Tweets with no links will get skipped, so if you're not a fan of chatty banter, this will appeal to you.
Nambu is a feature-rich Twitter app for Mac OS X that supports many social networks. It offers plenty of configuration options but has simple defaults for beginners. It also has excellent integration with its sister services, tr.im and pic.im
Twalala is a simple web application with some stream filtering so you can control what you see, and what you don't want to see. If their name confuses you, think of a kid sticking his fingers in his ears and saying "lalala!" to block out what someone else is saying. From their website:
Get ready to take control of your twitterstream. twalala is a client for Twitter that allows you to control what you see, and more importantly, what you don't see in your twitterstream...
Finally, Twitter with a mute button.
TweetDeck is an Adobe AIR Twitter app for power users. Using a column layout, TweetDeck allows a user to set up many feeds at once. Some are specialized to Twitter, such as your friends, @ mentions, direct messages and so forth. But the real power lies in their numerous advanced search features, which filter the public timeline any way you desire: