Class (originally called Grade) is an overall measure of how your monster is advancing in its training; think of it as moving through grades in school. Once your monster is strong enough to be considered in a higher class, it will move up. Do not confuse Class with Rank; the first refers to the monster's progress while the second refers to the breeder's progress.
Your class starts over again every time you raise a new monster, and if you're raising several at once they may all be in different classes.
The class system for the first four games and the two Advance games (In Monster Rancher Advance, there is no "E" class) are as follows:
In these games, you rise up in class according to how many Monster Association Official Tourneys you win. These specific Monster Battles are held every few months and are meant to assess whether or not your monster is ready to move up in class. When you do move up, the Association will send you a notice in the mail.
In these games your Class and Rank both run on the same letter scale but your Breeder Rank does not start over again every time you raise a new monster.
These games replace Class with something similar called Level. Level allows you to see a monster's overall potential at a glance. Level can be any number from 1-100.
In this game the class system is completely redone again. It is as follows:
F-1, F-2, F-3
E-1, E-2, E-3
D-1, D-2, D-3
C-1, C-2, C-3
B-1, B-2, B-3
A-1, A-2, A-3
S-1, S-2, S-3