I'm looking into dowloading LimeWire Basic onto desktop running Win XP. I get a window opened up asking to Run or Save the file/download. Which should I click on>?

Save the file. Then once the download finishes, double click to run the installer. It's always worth saving the installer, in case you ever want to reinstall that particular version again (just in case other versions don't work as well).

I save my installers to a folder in My Documents. It depends on where you have downoads set to save within your browser's settings. If you're downloading to desktop, then after you run the installer you can cut & paste that into My Documents or another location (desktop isn't always the safest place as it can sometimes be the first place to suffer if your computer gets infected, files may not be preserved if you have to do a system restore, files on the desktop could be easily deleted by mistake if somebody else uses your computer) & just leave the shortcut to LW on your desktop.

Hello! Thank You so much for your assistance with this. I will do just that. I agree with your idea of saving it to My Documents and the ease of someone deleting it from desktop. I'm not one for too many icons on there and such on there.
Once more, THANK YOU again.