1) AVI
AVI is a media container format like MOV. It is it self not the Video. The format of the media could be anything (MPEG, DV, Text etc). QuickTime does not support all types of media especially if they're wrapped by an AVI!
In most cases AVIs are some sort of DivX/MPEG4 Videos. DivX is not supported by QuickTime and although MPEG4 is supported, it is not if it comes as AVI. You must download a codec whichs adds support for these media types to QuickTime.
Try this one:
www.3ivx.com
2) MPG/MPEG
There are at least three MPEG-Video formats on the market. MPEG1, MPEG2 and MPEG4. All of them have the same extension (i.e. .mpg or .mpeg). QuickTime supports only MPEG1 and MPEG4! For MPEG2 you must buy an extra codec from Apple
http://www.apple.com/quicktime/products/mpeg2playback/
The Videos on DVDs are in MPEG2. I guess the Videos you've downloaded are also MPEG2...