When I started out with twitter, I used the native OSX client Twitterrific. It's a pretty nice piece of software, but I couldn't get comfortable with it popping up every time there was a new tweet, as it was a bit too disturbing, and if I turned it off, I forgot to check it out at all.
However, lately, I have a much better solution. I already communicate fairly frequently through the venerable Internet Relay Chat protocol (or IRC as most people know it). Thanks to semifor, I now have Twitter as just another channel in my irssi screen:
It runs as a separate process on my irc shell, pretending to be a IRC server. You can run it as a in the background, but for now I'm pretty happy with having it running as a new window in my screen session. Yes, I know that this setup marks me as a unrecoverable geek, but I don't care. It's just so sweet :-) If you don't have root privileges on your shell, you might find local::lib useful to install the required dependencies.