What are the different parts of a bike tire? (Of a regular bike, not a mountain bike!)?

That will be the rubber tire, the inner tube, and the wheel that the tire and inner tube mount on. The wheel is made up of the rim, the spokes, and the hub. I think you are meaning to ask how many different parts makes up the bike wheel/tire combination. Many people mistakenly talk about the wheel when they mean the tire. The actual tire is just a casing that protects the inner tube. It's the inner tube that holds the air. It's the inner tube that can be punctured and then the air escapes and that causes a flat tire. The inner tube is positioned inside the tire. The tire is made to fit on the rim of the wheel. There are many sizes of wheels and tire combinations.

Bead (wire bead or Kevlar), side-walls, tread, bias plies. You can also count: Rim strip, Inner Tube, Valve-stem.