It's about time one of these came along, and I'm happy to say that it's been done right. The app has five tabs: search, listings, saved listings, saved searches, and location. As you can probably guess from those headers, you can save listings and searches.
What you may not have been able to deduce is that you can set the app to recheck your searches in specified intervals. So, if you want a great deal on an iPhone, but they're just getting snatched up too quickly, have this program re-run the search over and over again. It will notify you when it finds a match.
Craigslist Checker currently supports read-only functionality in the United States, but both of these limitations are listed as possible upgrades.
At the time of this writing, the title of the app in misspelled in the Market. Search for "Craiglist Checker," (no s after Craig). More info can be found at the creator's humorous page.