I am not very Mac literate, but I am trying to install LimeWire on a Mac OS8.1 system, and all I get when I download it is a LimeWireMac.bin file. What do I do with this? Double-clicking it tries to open it in Acrobat Reader.

A .bin file is a MacBinary encoded file. You need Stuffit Expander or Internet Explorer 5 to decode the file so that it is usable. I'm not sure what version of Stuffit Expander and IE is even supported on (the rather ancient) OS 8.1. Check

http://www.alladinsys.com

for Stuffit Expander if you don't already have it.
Drop the .bin file on SE to open it.

Or in IE's preferences, under the Downloads option make sure the "Automatically decode MacBinary files" and "Automatically...Binhex files" options are checked.

Hope this helps...

No browser decodes .bin or .sit files, they simply direct them to Expander automatically.