Tires are the rubber part of the overall wheel, and are filled with pressurized air when properly in use. The part of the tire which is in contact with the ground is called the tread. The tire's edge is called the bead, which seals it against the rim of the wheel.
Tires wear down over time because the heat and friction scrape off minute amounts of the tread's rubber over time. The hotter the tires are, and the more they slide, the faster they wear down.
Uneven Tire Wear
Tires can wear unevenly (one side of the tread wearing more than the other side, or some areas wearing down faster than others) due to too much/not enough camber, driving aggressively, locking up the brakes, or poor tire design.
Tires come in many sizes, with both width, height, and profile varaibles.
Wider tires are favored for their increased grip, compared to thinner ones.
A taller tire is able to travel a further distance with one full rotation than a smaller one.
The profile of a tire is the difference between the overall height of the tire and the height of the rim. Low profile tires have a smaller sidewall, which doesn't give way as much as a taller sidewall.
The bead of the tire is the innermost section which sits on the rim, sealing the tire and keeping air from leaking out. If the bead is damaged, air can leak out (the leak's speed depends on the amount of damage). A rim with a sharp edge can cut the bead, and will keep doing so after new tires are put on.
Modern tires are made out of a rubber compounds with metal ply stips running though them.