With hundreds of add-on tools, Twitter certainly has plenty of ways you can analyze its data. I set out to find the best tools that I would recommend for you to track and compare your own Tweets, as well as examine the growth of followers and when you actually send out your 140-character missives. My two faves are TweetStats and Twittercounter. As you are resting from your Thanksgiving feast, you might want to try them out, along with several others that I will show you.
There are other tools that involve "sentiment analysis," being able to examine what people are Tweeting about or the attitude they are expressing in their tweets. The tools in this article are mainly for more quantifiable metrics. We look at whether you need to pre-authorize the tool to access your Twitter account, how much customization is available on the reports displayed, if there is a fee to use the service and whether you will need something outside of your browser to do the analysis. Most of these are completely free, which is nice since you can experiment and see what makes the most sense for you.
Some of these tools are dirt simple: you enter the Twitter ID or IDs of the appropriate people and wait for them to create their reports. Some of the reports are quite ugly and could use some design help, although as Joe Brockmeier points out, none can be as bad as the native Mac Twitter client: "it's like they got the UX designers that weren't quite good enough to design Windows ME on board."
You might also want to review an article that I wrote last month about 17 alternatives to Klout for other services that go beyond Twitter, or that attempt to measure some kind of reach or influence in social media.
The following tools are listed in the order of most to least useful, at least to my point of view. If I have omitted any of your favorites or have maligned or overhyped a tool that you use, please share your thoughts in the comments.