We have cats. Mod-80 mentioned some of the issues. Caring for animals is seen in the Jewish tradition as something that can inspire us to be more caring about our fellow humans. Cats and dogs, the most popular pets, have also provided direct human benefit: Cats because they control rodents (and indeed many small storeowners in NY continue to use them for this) and dogs for a wide variety of work.
Here are some more halachic issues:
While you can’t feed your pet a meat and milk mixture, you CAN feed it a poultry and milk mixture because the accepted halachah is that poultry is meat d’rabbanon.
You must always be sure your pet has food before you eat.
You can’t feed your pet chametz on pesach but you can feed it kitniyot even if you are Ashkenazic.
If your pet, without your prompting, uses its claws to cut toilet paper on Shabat, you can use the cut toilet paper. (I actually asked this shilah and that was the response I got!)
Cat litter is muktzah on Shabat; you can’t make blessings near the litter boxes.
Pikuach nefesh does not apply to animals.
Some communities have the minhag of not keeping non-kosher animals as pets; I’m not sure what is the source for this. This would eliminate cats and dogs, the two most popular pets.