Since you weren't specific as to whether you were referring to Halter or Performance classes, I'll mention the halter side of it too.(at least to the way I have always understood them)
FAULTS: This can be anything that does not conform to the Breed Standard of Perfection. So for example, cow hocks, ewe neck, weak hip, can all be considered faults. You can still show in the class(es) but if there are other horses with less faults, they may place over you, and possible even be the cause of your ELIMINATION.
ELIMINATION: This would be if there are 20 in a class, and only 2 are chosen in the end..Champion and Reserve Champion as an example. Because the judge(s) felt they were, in the judge's eye, closer to the Standard of Perfection, they stayed in and were placed in those two available positions, thus, ELIMINATING the remainder of the class, which would be 18 in this example, and those 2 would go on to compete further.
DISQUALIFICATION: This would be totally excused from the class for a reason of a disqualifying fault or behavior, as set out in the Breed Standard and/or Show/Association/Registry Rules. Examples may be cyptorchid in a mature stallion, bad bite, or ill mannered behavior while in the ring etc.