Why can't my computer play some .avi (or .mpg, .mov, or .qt, etc.) files? I get asked this question frequently by engineers and "normal" people at the grocery store. The answer is that digital movies have 1) a wrapper and 2) a CoDec. Your computer must understand both. Two common wrappers are (Video for Windows) AVI and (Quicktime) MOV - these determine all the controls you can manipulate as the "consumer" of the video. CoDec stands for Compressor/Decompressor, and the named codec algorithm (like mpeg 1, cinepak, DivX, etc.) defines how the images were stored internally: they are compressed to take less disk space, but often at the price of reduced quality.

So you downloaded an .avi file because you have Windows Media Player - will it play? The player can open the wrapper, yes, but you will only see the images if your computer also has the appropriate codec installed - some are so common that they were part of your system by default. Cinepak, for example, is fairly common, and Cinepak-compressed video can be wrapped inside a .avi OR a .mov; Two versions of the same movie can have almost identical sizes, since the only difference is a few bits in the file header. But Sorenson is a flavor of codec that only fits with Quicktime wrapping.

Further:
The above is true, but simplifies and does not address many other digital video practices. For example, one might think that the file "movie.mpg" would require a distinct player that understands the MPG wrapper, but MPEG 1 is actually a codec, with an intentionally agnostic wrapper, so many players can unwrap the data.

In my opinion: MPEG 1 is currently the "safe" way to distribute movies since almost every computer platform has a player that decodes .mpg's by default, but DivX can often pack more detail/quality into a smaller file size.