The tire is such a commonplace item -- it's on every car, every truck, every bicycle, every aircraft. It's easy to not give the tire a second thought, but like every other technology, the tire has an interesting history of advances and failures.
In the 19th century, carriages and wagons used steel strips for "tires" on their wheels, with the punishing sort of ride that you'd expect. In later years, they were shod with strips of natural rubber, which was an improvement but was still problematic. Solid rubber still rode pretty rough, and the natural, uncured rubber would get gummy in hot weather and shrink and harden in cold temperatures. Charles Goodyear was able to help with the invention of vulcanized rubber, but the modern tire was still several years off.
By the 1880s, the bicycle was becoming much more popular, and in 1888 Scottish engineer John Dunlop was watching his son struggle with the bone-shaking ride of his tricycle. He then devised the first-ever air-filled pneumatic tire, and a few years later Edouard Michelin developed the first "clincher" pneumatic tire, easily removable for repair.
These advances coincided with the development of the horseless carriage, of course, and soon speeds were starting to pick up and more was expected from tires. Things like inner tubes to hold air and grooved tread patterns for tires soon followed. By the 1910s, engineers were designing tires with angled layers of cotton cord beneath the rubber surface, adding durability and strength, and the bias-ply tire was born. Bias-ply tires would soon become the industry standard and would remain so until the 1960s.
The next big step forward in tire design was Michelin's radial tire, which featured steel belts and fabric plies that were set at a right angle to the tread instead of layers which crisscrossed at angles. Radials offered longer wear, better handling and road manners and soon became the standard in Europe, but they didn't really catch on in the US until the 1970s.
Tubeless tires debuted in the 1950s, and tire design continued to evolve with improved rubber formulations, better tread patterns and a variety of new tire designs such as the all-season tire, UHP tires, grand touring tires and other newer developments. Today, things like the run-flat tire and tire pressure monitoring systems have made tires more reliable, safe and long-lasting than ever before!