Everyone who writes about Twitter has to compare it to other things. For me, it's most like what we did in MUDs when I used to hang out there (a kind of chat world, see MUDs and MOOs on wikipedia, and my book about one). We used to connect while working, and "idle" much of the day, but "wake up" to post links to things we thought were interesting, or to say what we were doing "in real life." We even watched TV together in a MUD group. In a MOO, you had to do something special to direct comments to someone, just like you do in Twitter (where you prepend "@name"). Voila, c'est Twitter; except that in a MUD you had to go somewhere to be in the space by connecting specially, it was less public, and a lot more synchronous. Plus not searchable from "outside" the MUD client. So, okay, it had some differences.
Other things it's like: How people change their "status" message in a chat client, and sometimes riff off other people's status messages. That's not archived in the way Twitter history is, though. And it's like SMS, in that's it's terse, but for a party. And it's like a very slow chat room, where no one really knows who's listening in or who might look at what you said later. (Watch out.)
Brief geeky research aside: There's an old paper by Clark and Brennan (1991) that's a goodie among people who study CMC (computer-mediated communication) that describes potential aspects of communication media, including whether they offer co-presence, visibility, co-temporality, and sequentiality of messages. To really consider how Twitter stacks up, you would also want to consider system features that characterize rich Internet communication tools, such as the potential for users to have private one-to-one and multi-party conversations that aren't recorded, what kind of message size is possible, availability of threading/sorting/filtering tools, ability to archive exchanges and/or prevent it, possibility of editing posts after they are made, ability to block messages from certain people.
Twitter is less synchronous so less co-temporaneous than internet chat or face-to-face or phone talk, the reviewability is possible but only fair in practice (in that you have to do some work to go back in a history to check what you missed), and interruptions between two-person exchanges are common. Threads are possibly even impolite. Private messaging is possible depending on the client used. Editing isn't possible after posting, but deletion is. The message length constraint strongly restricts the type of exchange that can happen, by design. Blocking of a kind is possible. You can filter your list of followed people to a "favorites" list if you want.
Which reminds me - all communication media allow for genres or registers of speech/writing, in which the style and topics can differ tremendously across groups of users and occasions of use. Generalizations about how people use Twitter will only be applicable to local groups of followers and their following. So I won't try. Give a look in and see what you think.
Because of the very public nature of Twitter, we get the possibility of search tools like Summize. Which means you can look up keywords or people and find out what's up with them. You can even subscribe to these searches by RSS, so thatt you can follow public chat that mentions your favorite product or TV show. (When I mentioned FIOS once, someone who works at Verizon started "following" my comments on Twitter.)
Summize also allows for interesting meta-search applications like Twitter Spectrum, allowing you to contrast word environments for two terms. Just for fun, here's a few charts of contrasts I find interesting. You can see who's talking more about what here.
Anywho, I'm enjoying Tweeting, although I still miss ElseMOO after all these years.