If you've ever used Twitter on Android, you know two things - one, there are a lot of apps available, and two, none of them do everything you want. I'll try to help you get as close to the bulls-eye as possible.
So you want to know who ragequit you after you posted pictures of your brunch? This is your app. Lots of good analytics and tracking of account details; just don't say I didn't warn you about obsessive behavior regarding your follower count.
twicca is free. REALLY free, not bait-and-switch-ware like Seesmic, Tweetcaster, Plume, and others. It may not be as polished as those apps, but it makes up for it in function and expandability. Power users should look no further.
Janetter first caught my eye as a Mac app. The Android version is almost as good - and there's a bit of TweetDeck spirit in it, for those who lament that app's demise. A contender against the "usual suspects" in Android Twitter apps.
Hootsuite is THE social media app for corporate teams. Power users may like its feature set but it shines brightest for multi-user/multi-network teams. If you have many people working on a few accounts, this is your app.
Some people are going to be satisfied with the "official" app, and you may be one of them. There definitely are apps out there with more robust features, so if you are a power user, check those out first. But this one is good for no-frills access.