For most games, it means quicker loading times and less noise from the DVD drive. Could potentially mean less wear and tear on your game discs, and if they do get a little scratched, they'll probably still work since the data is backed up and the console only has to make sure the disc is present.
There are a few games where installing can cause problems or slow it down. Blur has audio problems when you install it. Halo 3 wasn't designed to be run from the hard drive and Bungie have said it could possibly make loading times slower, so don't install that one. Most newer games should work better if they're installed, though, since the feature was added in 2008.