There are a number of ways you can get your specifics. The easiest is to go to the Apple menu and choose "About This Mac"
That will give you the type of Mac, the memory installed, and the version of the system software. There is also an option to gather much more detail--like the kind you need to see what kind of CD burner you have.

If you have a new Mac, spend some time looking at the System Preferences, especially the Software Update (run the update) and Sharing items (find the firewall tab, click on the New button, and look through the list to find the item that says "Gnutella/Limewire (6346)).

Basic troubleshooting is as simple as running a "Safe Boot" and "Reparing Permissions" (see http://www.macattorney.com/tutorial.html ).

For more details about LimeWire, look at the Prefereces and find the item that says "Bug Reports." Click on the button for "Sample Bug Reports", select all the info and copy, or just scroll to see what kind of information the developers use to track down problems--most of it won't make much sense to average users like us.

There are other filesharing tools for OSX users, but LW is the best I've tried. Filesharing is very complex and still developing quickly. As you saw the first time you tried it, it can work right out of the box, but I suspect other hosts either "evaluate" new members, or just happen to have open slots when you tried to connect. Take a look at http://www.limewire.com/english/content/netsize.shtml to see the best times to connect.

Hope that helps you get going on figuring out how to connect. Do remember to post back and help others too.