twitter is a great way to post little snippets of information, or teasers, to let friends, followers and the world know what you're up to, thinking about, or finding interesting at the moment. The salient feature of a twitter message (tweet) is that it is short: there is a 140 character limit. This removes a barrier to getting messages out - one doesn't have to craft beautiful, reflective prose entries. Tweets are more about stream of consciousness and what's going on in the moment.
I use twitter as a learning tool: I finding and following people I think are doing interesting things in mathematics, and education. Following their tweets helps me to figure out what is interesting, important, new, popular, etc. in a timely way. People I follow may be colleagues, mentors, innovators, students, or perhaps not easily characterizable.
Twitter can also be a teaching tool - a way to get those brief flashes of observation, insight, reflection, inspiration out there quickly, where they can be noticed, appreciated, reflected and commented on, and otherwise 'processed' quickly, or recorded for later searches.
I'm setting up twitter integration with my twitter account @mathseeker and this site - basically, any blog post I post here should get displayed as a tweet from my twitter account, and my tweets from this account are automatically rounded up and posted on this site.