Twitter is a stand-alone microblogging service -- you don't have to have an LJ to use it. You make teeeeenytiny posts to it. You can do this from their website, from text message, or from a dizzying variety of clients. Twitter was not originally intended to be fed into LJ, but people are just crazy like that, so naturally things evolved.
In common use, stuff gets posted to Twitter that one might never consider making a full LJ post about -- location of the moment, weather, fleeting moods, passing thoughts, and daily trivia. museumfreak sums it up quite accurately:
The original function of LJ was to answer the question "What are you doing today?" Twitter answers the question "What are you doing right now?" The former is of course the integrand of the latter. I post these to LJ because I want to archive them better and I want my LJ friends to know what's going on with me even when I don't have time/energy to post.
I find that since starting to use Twitter, I post much of the same content to Twitter that I would ordinarily save up for a link soup post, and the same not-topic-focused thoughts-of-the-moment that I would usually unload irrelevantly to whatever IRC channel I happen to be in. It's not often important enough for its own post, but put it all together and you have a post. However, since I am a feedback junkie, I want to get these thoughts out there NOW to my people, not wait for enough content, or focus long enough, to have an actual LJ post. It's only after tumbling things around in my mind for a while that there's enough coherent to create a real post.
IRC is designed to support multiple channels on multiple focused topics, even though we stray from them and stray from them badly. Since a lot of the usual suspects on IRC are now also on Twitter, the general focus in the usual channel has improved, while conversation has dropped off somewhat. IRC will generate side-channels to support conversations that the general channel isn't interested in following, although when moving to a side-channel a lot of the conversational momentum gets lost in ensuring that everyone is on board.
IRC is for my topics. Twitter is for my people.
Granted, a lot of my people on Twitter are also my people from IRC, and I don't think that's a coincidence. In turn, IRC has pretty much replaced most of the single-person chatting I used to do (there are exceptions).
How To Use It (technically), and Stuff You Should Know
To get started, first you sign up for twitter here: http://twitter.com
Figure out whether you're going to want it to feed into LJ or not. If you are, find a service that will do that. One of the most popular is LoudTwitter: http://www.loudtwitter.com/ -- but you can also run a script on your own server if you prefer that, and have a server available.
You can choose to make your Twitter updates Twitter-friends-only, however, doing that generally interferes with your ability to post them to your LJ unless you're doing something really fun with your scripting.
If you don't mind a public Twitter, but your journal is friends-only, you can post stuff from Twitter friends-only; the easiest way that I know of to do that is by setting your minimum security, which you should do if you have a friends-only journal anyway.
Regardless of your settings, you then update from Twitter, and make sure that stuff is coming through when scheduled to (sometimes Twitter, or LoudTwitter, or both, are flaky).
People on your friendslist who also use Twitter will notice that you are now doing this, and either add you on Twitter (or not). Socially, some people may choose to not follow you on Twitter on the grounds that they will be reading it all on LJ anyway. On Twitter, it's called "following" rather than "friending". You can also block someone from following you, unlike LJ. Like LJ, however, public is still public.
The convention for Twitter usernames is to preface the username with the @ symbol. Twitter will autolink this for you. If you want to flag someone down on Twitter (comment to them), start a tweet with the @ symbol and their username. They will see it if they check their replies. (If you just put @whoever in the body of it, even if it is the second thing like there are 2 people, they won't necessarily get it, even though it autolinks. Them's the breaks. Other things that don't work: @ whoever, because they're not touching.) Some people also don't check their replies on Twitter, so it's not safe to assume that someone has actually seen a Twitter-reply. Some people choose not to watch the @somebodyelse updates in their Twitter stream, so don't assume someone has seen something if you've put it in a message to someone else, even if they're watching you and it's public (unless, you know, you know otherwise).
You can get the firehose of twitter-friendslist via text message, nothing via text message, or many settings in between.
Twitter allowed subscription to certain words. This isn't available via the web right now; I'm not sure if it's still available via SMS. Stuff has grown up around that, though, to allow somewhat of the same function via different routes. If you are making an announcement to the world on a topic, you can include the symbol # in front of the name of the topic to make it easily searchable. Optionally, add @hashtags as a friend. This was notably pioneered during a 2007 California fire, with tweets including the keyword #sandiegofire. The syntax is reminiscent of IRC channel definitions, but can be used anywhere in the tweet.
Some commentaries and utilities:
http://twitter.pbwiki.com/ -- huge wiki of Twitter-related stuff
http://a.wholelottanothing.org/2008/11/why-im-blocking-you-on-twitter.html -- good rules of thumb for dealing with spammers and the like
http://paulstamatiou.com/2007/01/26/stammy-script-rss-to-twitter-using-php -- syndicate stuff (from LJ) to a Twitter account. (Note: I do not recommend syndicating stuff from your LJ to your Twitter, and then importing your Twitter right back into your LJ, because the scripts will keep each other going until something dies, albeit on a daily basis and not realtime like an email loop.)
http://tweetscan.com/ -- search public Twitter and other microblogs
http://www.twittermail.com/ -- post by email (why?!)
http://twittervision.com/ -- view random recent tweets mapped
http://www.hashtags.org/ -- search opted-in twitter accounts (people following @hashtags) for topics marked with #topic
http://twemes.com -- hashtag search without opt-in
http://twitterless.com/ -- track twitter defriending
http://userscripts.org/scripts/show/17139 -- LJ Twitterless Greasemonkey script for stripping LoudTwitter posts from your friendspage
http://flourish.livejournal.com/253762.html -- excellent introduction
http://jott.com -- transcribe voice stuff and send it somewhere (like twitter. or LJ.)
http://ping.fm -- update multiple places at once