A counter is when you hit the opponent while they are trying to attack you. You will do this all the time when fighting the computer in the various modes.
A Fatal Counter is when you hit with a counter, but use a specific attack that causes a Fatal Counter. Every character has different Fatal Counter attacks.
I used Relius Clover who has a Fatal Counter of forward + C. I got this naturally while doing the other modes but you can easily set it up in Vs mode with two controllers.