Arizona Law (ARS §28-101) defines All-terrain vehicles as a motor vehicle that are all of the following:

(a) Is designed primarily for recreational nonhighway all-terrain travel.
(b) Is fifty or fewer inches in width.
(c) Has an unladen weight of eight hundred pounds or less.
(d) Travels on three or more low pressure tires.
(e) Has a seat to be straddled by the operator and handlebars for steering control.
(f) Is operated on a public highway.

There is specific equipment (ARS §28-964) that ATVs must have regardless. If you are under 18, you must wear a helmet and goggles at all times. It must be equipped with a rearview mirror, a headlamp (ARS §28-924) seat and footrests for the person driving and any passengers (ARS §28-892). Handlebars must be positioned so that the hands of the person driving are not above shoulder height when sitting on the seat. An ATV driven on a highway must be equipped with at least one brake operated by hand or foot (ARS §28-952) which meets specific requirements for braking distance.

You should call the local police to see what other specific ATV laws your city might enforce on city streets.