Tires are the rubber part of the overall wheel, and are filled with pressurized air when 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.
How to Read a Tire's Sidewall
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. Most tires have a code on them to determine various sizing qualities.
Wider tires are favored for their increased grip, compared to thinner ones, but very wide tires have cornering traction downsides as well. If a vehicle with super-wide tires drives around a sharp enough corner, the side of the tire closer to the inside of the corner travels a shorter distance than the side of the tire on the outside of the corner, and part of the tread has to slip or else the tire will twist.
A taller tire is able to travel a further distance with one full rotation than a smaller one. The rim size is not the tire's overall size, but it is a limiting factor in how small a tire can be.
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. A bent rim can also prevent a bead from sealing all the way around the rim.
Modern tires are made out of a rubber compounds with metal ply stips running though them. Radial ply tires are the most common type found on modern passenger vehicles.