To admit, I don't use a Twitter client on my desktop computer. This may sound dumb, however, I'd rather stick to a browser and the Twitter web site as is. Generally, I'd rather have everything in browser... well maybe some day.
However, when it comes to iPhone, I really like specialized applications. For example, Facebook has a really cool iPhone version of the site for the Safari browser. Yet, I prefer the standalone application.
I'd been using Twittelator for accessing Twitter from my iPhone for months. I really liked Twittelator, however, it does have some annoyances (well not precisely annoyances but lack of functionality I'd say). But Twittelator is FREE, which makes it a very good and decent app. And here comes TweetDeck. Which is also free and which is a far better application (this is my very personal opinion and I did not intend to offend Twittelator developers in any way :). TweetDeck belongs to a commercial-level software and it's so cool to see this app free.
As I have mentioned above, I don't use any Twitter clients on my desktop, however I've heard that TweetDeck is one of the most popular desktop clients for Twitter.
So, if you have not downloaded TweetDeck for your iPhone, I highly recommend this application.