There are a couple of resources you can use. I've returned several class rings and it was pretty easy and I have one that I've been trying to return for years and no luck.
First, go to www.google.com and search up the school. Now find the "contact us" area and sens an email to (or call them, the number is usually there) and explain who you are and what you have. Unless the ring is decades old, they can usually take the class, initials, and any other markings and find the person. Typically they will take your name, address, and phone number (or whatever you're comfortable giving them) and tell you they will have the person contact you when they get hold of them.
The other thing you can do is to go to http://www.classringfinder.com/ where you will find a "Lost Rings" and a "Found Rings" section. You can add a description of the ring and a picture there. Here's one that I put up there some years ago:
http://www.classringfinder.com/RingDetail.asp?RingID=1216
There is a clearly identifiable nickname that is not visible in the picture that only the owner would know, so no way somebody could "identify it" from the picture.
Anyway, that's all there is to it...pretty simple.