When you think about actually making two computers (like maybe the big processor inside your PC and a small microcontroller inside a USB peripheral) communicate with each other, the first problem you have to face is:

What are you going to have them say to each other?

There's a big fancy name for the solution to that problem: it's called a protocol.  It's sort of like a script: what is each side going to say to the other side, and when?

A USB Class is sort of like a protocol: it's a collection of rules that govern how the Host and the Peripheral device will talk to each other, and what they will say under what circumstances.