I bought a basic female to female USB adapter and connected my Galaxy SIII USB cable to the mouse. This didn't work. What am I doing wrong? Do I need an OTG cable? Something about USB hosts? I don't understand how to get this to work.
Yes, you need an OTG cable. The pin layout is different than a regular micro USB to USB cable; essentially, it's equivalent to shorting pins 4 and 5 on a normal cable. This is what causes the phone to know that it needs to be the host rather than the slave.
In fact, you can do this yourself if you're handy with electronics and have a cable you don't mind taking apart, though they're also readily available at many online retailers (Amazon, Newegg, Monoprice, etc).
To connect any device to your Galaxy S3 (or any other host-capable Android device), you need a OTG host cable. Without that host cable, the phone doesn't know it's supposed to act as the host rather than as a peripheral.
This is an example of such a cable. Most online retailers carry similar ones.