Twitter was designed for Text Messaging
I find it ironic that people forget that Twitter was originally designed as a way to create groups for text messaging. After all, where did you think that 140 character limit came from? SMS has a 160 limit, and they reserved 20 characters for Twitter to add instructions or labels to the message.
So, rather than texting a whole bunch of your friends individually, you send a text message, then twitter relays it to their phones, and they can decide whether or not to get those text messages.
You send your updates to Twitter by texting to 40404 (in the US). The updates show up online just like a posting from any other Twitter client. This is handy if you have an old phone without a Twitter client or you just don't want to run, log in, and deal with a Twitter client or web on your phone.
The feature is still there, and you can choose to turn it on with a few simple steps:
1) Log into Twitter on the web
2) Click on Settings (just like if you were going to change your profile settings)
3) Click on Devices, add your phone number.
4) Got to notices and you can turn on DMs to your phone.
5) To get updates from individuals texted to you, go to their profile and click that little mobile icon. (This would get really annoying if you turned on updates for everyone you follow if you follow thousands of people).
You'll be given instructions for how to verify your phone number before you start receiving texts from Twitter, but it's pretty simple.
Of course, there are other text message services like Twitpic or Facebook Mobile, both of which I've been known to use, but not many of them actually text YOU when something happens on your account, and that's one way Twitter is different.