From Diablo Wiki
Pets of any sort have very varied attributes. In some cases it's as simple as a small creatures sent in large numbers to die horribly while dealing some damage, and it can be a summoned beast that is even more dangerous than the original caster.
A "pet class" in terms of character class is a class that is not only capable of summoning a minion, but is also to some extent dependant on this summoned creature. They would be slightly more frail and deal slightly less damage than other classes in exchange for having a separate unit helping them killing monsters.