First, there's already a sticky about this:
http://www.akiba-online.com/forum/showthread.php?t=208557
Now, I think you need to be a little careful when dealing with avi files because of the avi container and other issues if the avi files are from different sources.
If the avi file was originally split using a simple file splitter, then you should be able to combine them back to the original by simply concatenating them together using either of the two programs already mentioned, although the simplest way imo would be via command line if you are comfortable with it, since you won't have to install any additional software. You can simply open a command prompt in Windows (or a terminal in Mac), go to the directory where your files are stored, and type:
"cat part1.avi part2.avi part3.avi > complete.avi"
However, if the avi files are from different sources, or even rips from the same DVD that were encoded into two different files with the same settings, then I'd recommend using one of the simplest and best video editing tools around-- VirtualDub. You can find lots of tutorials online, such as this one:
http://www.afterdawn.com/guides/archive/virtualdub_join_avi.cfm