Follow me. The highlight of Twitter is that you can use it to "follow" the posts of anyone you want including the companies and people you're targeting. Many viewers follow me (@ToryJohnson) on Twitter to ask career-related questions and, because of the length, I can quickly respond to everyone. I also post job leads as I get them, which is another reason people opt to follow me.
Follow your next employer. Let's say you're set on working for a specific company -- take Whole Foods, which has a very popular Twitter following. You can read what customers and employees have to say just about every minute. And if you're trying to interview there, it helps to understand the culture and the dynamic among its fans and foes. Many companies also post their job openings on Twitter, which is another reason to follow.
Twitter tools. One of my favorite sites is search.twitter.com, which enables you to search by key word, name, employer or buzz phrase to see what's happening at that moment on Twitter.
TweetMyJobs is a tool that enables you to search for specific openings by industry, company, key word, etc.
A word of caution. Everything you post becomes public and there are many infamous anecdotes about a candidate getting a job offer, then posting a comment online saying, "Yeah I got the offer, but the commute really stinks" or "I got the job but the money's pathetic." Assume it'll get back to the prospective employer and the offer will be rescinded before you even start.