Mac OS X Shutdown Issues

My Macintosh running OS X has been rock solid for a year. Recently, I’ve noticed some very peculiar behavior. The screen gets a little wacky. My background image gets fuzzy in spots. Windows that I close leave remnants on top of other windows. Little dots appear around my pointer. Eventually (normally sooner than later), my menu bar at the top of the screen flashes red. A shade drops from the top of the screen to the bottom. And then a window pops up telling me to restart in a dozen different languages. Sometimes I can go all day with it happening only once. Sometimes it happens every 10 minutes.

I’m running Mac OS 10.2 on a 15″ 766 Mhz iMac. Has anybody else run into this? Does anybody have any ideas?