First of all, welcome to Apple's discussion forums and welcome to Mac. This is the best place to come for answers to any Mac question you may have. There are always tons of people here willing to help. You'll discover the Mac community is very helpful and we're very dedicated to helping each other.
Second of all, stop trying to install it. AOL's AIM client is very bad. I recommend either using iChat or Adium instead. iChat you already have (in your Applications folder), Adium is free, open source, and very flexible. It's the ultimate chat client and can do something like 11 different chat protocols (in addition to AIM). It's awesome!
http://adiumx.com
AOL's AIM is poorly written, it uses way too much memory, and it can cause all sort of problems on Macs and PCs alike.
.DMG files should mount like a temporary hard drive. .DMG means it's a Disk Image. If you open Adium's .DMG file, you'll get what looks like a hard drive on the desktop and in Finder. It will contain the Adium application which you should drag into your Applications folder. Then you can eject this temporary hard drive and open Adium from your Applications folder (but only after the file is done copying).
Post any problems you may have with installation here.
