The cemeteery in Appaloosa Plains has a nice sections set aside for pets.
You can set the life span for dogs, cats, horses, and Sim separately in the Options menu, and set a time span for each age group. Still, unless you turn aging off entirely, your pet will age up, and eventually die.
Aging up is a cause for celebration. You get a warning that the pet’s birthday is approaching. When the aging up actually happens, Sims will gather around and cheer, while the pet disappears in a flurry of sparkling effects, and the aged up pet appears.