There are a plethera of tires.

There are folding tires. These have a kevlar bead which means you can fold them up. Other tires have a steel wire bead which means folding them up takes some doing. Kevlar bead tires are a bit lighter. Often the same tire comes in both forms with the wire bead being cheaper.

On road wheels, tires range in size from 20 mm, 23 mm, 25 mm, 28 mm and 32 mm. Oddly enough, tire manufacturers can not agree on just what a mm is so tires of the same "size" often are not. The thing is that the smaller the tire, the smaller the air pocket. That means in order to sustain your weight, the tire has to be pumped up to a higher inflation if it is a smaller tire. A larger tire will generally give a less jarring ride. It will also be more resistant to pinch flats when you hit a stone or pot hole. A pinch flat is when the tire is pinched against the rim causing a "snake bite" on the tube. It is opposed to the other kind of flat which is a puncture. It also seems to me that larger tires wear a bit longer. But they are heavier.

The final thing with tires is attempting to go with puncture resistance. Tires have various kinds of belts under the rubber to resist puncture. Some people swear by them. I have had poor luck with them but that may just have been bad luck. Tires only last so long. When they start to wear thin, any tire will be prone to punctures.

On road tires, you will often see all sorts of different tire treads and even no tread. It does not seem to matter in my experience. The tread is probably too small to really have much effect.