DivX is a video codex released by a company call DivX. Xvid is a free alternative to DivX, it's open source and totally free. Their filename extensions, (the something.xyz) is usually .divx or .xvid, though sometimes they are incorrectly labeled .avi
In someways it's like the differences between a brand name product and the generic copycat. Like Froot Loops vs Fruit Rings.
Wikipedia has good articles on both video formats. If you want to play them, the best program is VLC Player: http://www.videolan.org/vlc/
Well, many people seem to think that XviD is much better. Many people agree that XviD has much better quality and loading speed than DivX. But the only thing I know is that XviD is open source, and DivX isn't. But my computer works perfectly fine with DivX, so I have no argument here.
However, someone on Yahoo answers already asked this question, and it has been resolved. I think that question might be able to give you more info than I am. Here's the link: http://answers.yahoo.com/question/index?qid=20080622062018AA3zoIi