Like most mobile platforms, Android has no shortage of Twitter related applications. There is one, however, that stands above the rest: I Tweet! from Multiple Facets.
I Tweet! has all of the basic functions you would expect from a Twitter app. You can view all your friends’ tweets, people who @reply you, your DM’s, and send tweets (at this time, when publishing a tweet, Twitter says it’s from the API, not from I Tweet!). You can also view information from users’ profiles (name website, location, etc), and choose to follow or unfollow them.

There are several other things that make this a great application. First, there’s the ability to search, and it will even search in a certain physical location nearby to you.
There are a multitude of settings to mess around with, as well. You can pick one of two themes (I prefer dark), how often I Tweet! checks for tweets, replies, and DM’s, and even what color your LED blinks when you receive notifications.