AVI is a compressed standard, but I never personally liked the quality of most AVI's. I think MPEG is nicer.
.AVI doesn't have anything to do with compression -- it's merely a container format. The actual movie compression, features and anything else are 100% dependent on the codec used to do the compression.
The 'standard' codecs that come with windows aren't very good at all, but some external ones like Divx (kind of on its way out) and XViD are pretty nice.
So... when using AVI you have an additional hurdle in that the target machine also needs to have the codec on there that was used to compress the video file... Without the codec the video can't be rendered.
The advantrage of MPEG is that pretty much any machine is mpeg-capable out of the box.