I have an old computer in my house.

Windows 7 (32-bit)
1.5GB of RAM (some is taken up for graphics)
Intel Pentium 3 (3.0Ghz)

It is really slow. If you play youtube in full screen the video card crashes because it can't handle it. It has gotten so slow that you basically can't do anything productively on it. It is only used as a print server. A non-wireless printer is connected to it and it shares the printer via a windows homegroup so others can print wirelessly. I also have a program that broadcasts an AirPrint signal so I can print from my Mac. I want to install a lightweight Linux distro on it, but I am not sure how I would be able to have Windows and Mac computers wirelessly print. Does anyone know how I could do that? Thanks in advance!

For just a printer server there really is no need for a desktop and I would suggest running debian. The print server is called cups which is compatible with Mac computers and you can install software to make it an airprint server for Iphones, Ipads etc.

linux can share files and printers with windows PCs but it isn't compatible with homegroup. You will need to reconfigure your windows PCs to share using XP compability mode. You can print directly to cups without using windows file and printer sharing too.

I do have an old pentium 4 2.8 GHz with 768 MB of ram and using it as a print server like how you wanted. I am using puppy linux precise 5.7.1 and it's working flawlessly.

So you having a faster pc and with more ram, you will be surprise on how fast it will run on puppy linux.