Pet
If you have The Sims Unleashed (for the original) or The Sims 2 Pets (for sims 2) you can own a variety of pets and customize them for your leisure. There are pets in THe Urbz but those aren't as entertaining because all they do is piss.
The Sims Unleashed
The Sims 2 Pets
In The Sims 2 Pets you can have not only small animals (womrats, birds and fish), but a dog or a cat either! These are highly variable (you can set their 'personality' as well as their fur color and style), they have their own motives, learned behaviors, and will, and you can teach them certain commands. They even have their icon near your Sims'! But you cannot control them.