In Rift pets seem to come in two flavors: summoned pets for certain classes (beastmasters, druids, elementalists, necromancers, rangers, and warlocks) that have combat skills (these pets level up to 8 and can be trained further) and companion (or vanity or mini) pets that just follow your character around which can be purchased from the faction reputation vendors.
A player's list of various pets and where to get them.
Here is a list of companion pets. They can be killed but summoned back from the dead. Note there are also pets exclusive to various chain stores that sold the game. Walmart had the Shambler Hatchling, D2D had the Corpus Kitty, and Gamestop had the Bogling Wastrel.
Photos of various critters and information about where to find these vanity companion pets.