The official Twitter app is good, don't get us wrong, but a growing number of people think Tweetbot is better.
Why? Well, it makes it easy to follow tweets from specific lists of people for power users of the micro-blogging site, swiping through conversation threads is great and generally making sense of your never-ending tweetstream is just simpler and less of a headache with its well-thought out UI.
There's the usual push notifications and mute filters also mean you can block messages from users without unfollowing them. Sold? Oh, and you can save to the likes of Pocket and Instapaper. Done deal.