Actually putting all the parts of a tire together can be a messy process.
Rubber either coats or is the primary material in most of the parts of a tire. But the rubber used in tires isn't pure rubber; it's a mix of rubber (both natural and synthetic), oils and additives. Before a tire is built, all those components for the various rubber parts of the tire are mixed together in a banbury mixer, until they have the consistency of gum. From there, the material is sent to other machines, where it's fabricated into the individual parts of a tire.
Once all the tire parts are made, they get sent to the tire building machine -- which is a pretty apt name. On the machine goes an inner liner of rubber, the plies, the belts, the bead, the sidewall and the tread. Once all the parts are in place, the tire building machine presses them together.
The tire isn't finished yet, however. When a tire comes off the tire building machine, it's what's called a "green" tire. The materials have been put together, but they haven't cured. Plus, green tires don't have the grooved pattern of the tread in them yet.
Green tires go into a mold, where they're inflated. Inflating the tire causes it to press against the mold. As a result, the mold imprints the tread pattern and tire information on the tire. While in the mold, the tire is heated to over 300 degrees Fahrenheit (149 degrees Celsius) to cure the rubber and bond the components. It stays over that temperature for about 15 minutes for passenger car tires -- bigger tires, like truck tires, may cure for an entire day. The heating process is known as vulcanizing the tire.
From there, the tire gets inspected. Then it's ready to go on your wheels.