I used k-means clustering, a machine learning algorithm, to group tweets from the Obama administration. As this is an unsupservised learning algorithm, I had next to no control over how the clusters would form, aside from some basic normalization (removing URL's, replacing hashtags with words, stemming, etc).
Once I had these clusters, I gave them group titles based on what I observed to be the most common concept.
I visualized the results using d3. You can see the work at http://lieu.io/whitehouse-tweet-topics/.
The raw data dumps from Twitter are available in the repo here. The scripts I ultimately used to do the classification, in addition to a few dead-ends, are here.
I'm an engineer and designer working at DevelopmentSeed in Washington DC. You can find me on Twitter.