When he invented rubber tires in 1888 to cushion the ride of his son's tricycle he called them tyres. Perhaps to differentiate them from the leather tires already being made and marketed, not very successfully, by a man named Robert Thompson. But, more probably, just because he was from Scotland, and they like to spell things with a "y". The Oxford Dictionary of the English Language says the word tire is short for attire, and refers to something added around a wheel (i.e. "worn" by the wheel). The first tires where iron rims added to wagon wheels to make them last longer. Rubber tires filled with air were not practical until after Charles Goodyear invented the process of vulcanization (1839).
Soon after Dunlop's son got a smoother ride for his tricycle, pneumatic rubber tires where on various other kinds of vehicles, and the technology was there in time for the early internal combustion automobiles. Dunlop, however, marketed his tires mostly for bicycles. The company he founded is still in business, and is now a subsidiary of Goodyear Corporation. A corporation which bears the name of Charles Goodyear, although he personally never really profited from his invention his son was a somewhat better businessman.
Tires cushion the ride, absorb shocks, and also provide traction for wheeled vehicles. Grooved treads provide better traction on wet surfaces by giving the water a space to be squeezed into by the weight of the vehicle, but on a dry surface they actually reduce traction by reducing the amount of friction surface in contact with the road, that's why race cars use "slicks" with no treads.
Radial and steel belted tires were invented in the mid-20th Century, and have reinforcing bands built into the rubber which prevent flats and give tires a longer life. Most Modern tires are built from synthetic rubbers made from petroleum by-products, although for a long time natural rubber was an expensive commodity. Synthetic rubber was developed during World War II when America found itself cut off from most areas of natural supply.
Bicycles, cars, and other vehicles are made much more comfortable and useful by the use of tires, and we have Goodyear and Dunlop to thank for that.