Chapter 6 of the Houston Municipal Code is the Section that covers animals. Here are a few quotes from there:
Domestic animal means any beast or bird in common use that has been bred or trained to need and accept the care of humans and to live in a tame condition, including, but not limited to, livestock, goat, swine, dog, cat, fowl, or other beast or bird.
Sec. 6-4. - Running at large of domestic animals or fowl prohibited.
(a)The running at large of domestic animals or domestic fowl is hereby declared to be a nuisance. It is unlawful for an owner to permit any animal or fowl to run at large.
(b)It shall be a defense to prosecution under subsection
(a) above that the animal was:
(1)A dog in an off-leash site established under section 32-11 of this Code; or
(2)A feral cat that is a part of a trap, neuter, and return program.
So if you live in Houston, and your pet cat Fluffy is not a feral cat that was trapped, neutered, and released, then you are a criminal if you don't keep it on a leash outside.