For example, if you type something in upper-right google/yahoo search box in firefox there will be some kind ‘suggested auto complete’ sort of thing.
Another example is in youtube search box and Stackoverflow tags edit box just below this question preview. How do they work? What technology behind ’em?
That’s done with the use of AJAX, this site has a nice tutorial on it: AJAX Suggest Tutorial, and the WaybackMachine version, as website seems down.
A database with keywords and a bit of code is all there is to it as far as I know.
I’m learning how to use it right now actually, for work. 🙂
Another resource is w3schools. They have covered it as well.