Problems with Quicktime - need to reinstall?

That's the generic "application" icon; if you're seeing that on QuickTime Player X (I'm assuming you're running 10.6?) with that icon, it could be a sign of something more serious wrong.

Before you do anything else, open up Disk Utility in the Utilities folder in your Applications folder, select your hard drive from the list at left, and click "Verify Disk." Assuming it completes and says no problems are found, do the same thing for "Verify Disk Permissions." Once that has finished (it will almost guaranteed show at least a few things, most of which are trivial) reboot for good measure and see if that made any difference.

If the Verify step above shows errors, you've got some hard drive corruption; find the disc that came with the computer, start up using it, and run Disk Utility from the OSX installer menu there to do "Repair Disk"; hopefully that will fix the problems. Once you're done with that, restart and eject the install disc, then continue with the Permissions check above (don't do that one while you're booted from the install disc).

Assuming none of the above helps or comes up with anything weird, the possibilities are that you installed a 3rd party utility or modification of some kind that messed up Quicktime, or just that something went haywire with it. Unfortunately in the latter case, Quicktime is a deeply integrated part of MacOSX, so it's not that easy to reinstall it alone. You can try the QuickTime installer from Apple's website, but you may have to reinstall the whole OS (which won't require you to erase any of your settings, so long as you select the correct reinstall option, though you should have a backup).

Before doing any of that, though, have you installed any modifications to the computer recently?

Also: When you say that it won't play AVIs, do you mean that it gives an error when you try to open any AVI file, or just that some AVIs don't show any video? In the latter case, installing the free Perian QuickTime plugin (if you haven't already) will allow QuickTime to play pretty much any video file.