Within World of Warcraft the term pets is used for a variety of different small animals and other companions. Most commonly the term describes a hunter's companion. These hunter pets can be tamed from level 10 on and the hunter is able to command the pet to do various tasks, and he is even able to take direct control of the pet. For these reasons the hunter class is also called a pet class.
The other pet class is the warlock. While the hunter is able to tame any wild animal that roams the lands of Azeroth, the warlock is more limited in his pet choice. He learns the ability to summon demon companions or pets to his side, which will do his bidding. Depending on the talent spec of the warlock he will have four or five different demons to chose from. Additionally he can summon a temporary but very powerful pet.
Deathknights and Mages are technically not a pet class, but depending on spec they can summon a pet as well, supplementing their fighting power.

The other context, in which the term pet is used, is the so called vanity pets or companions. These non-combat pets can be summoned by any player and accompany them on their journeys. Although a player can only have one vanity pet summoned at any time, many players are trying to get as many pets as possible. Especially the rarer pets, which can only be attained from certain achievements or looted from difficult bosses, are highly sought after.