Early cars used wooden wheels, adapted from horse-pulled wagons. Some had metal rims for durability, or solid rubber treads for improved traction.
Charles Goodyear invented vulcanized rubber in 1844 that was later used for tires.
Robert William Thomson invented the first vulcanized rubber pneumatic (air-filled) tire, patented in 1845. His invention worked well but was to costly to catch on.
In 1888 a patent was granted to Scottish veterinarian John Dunlop for a pneumatic bicycle tire.
In 1895, André Michelin was the first person to use pneumatic tires on an automobile, but not successfully.
In 1911, Philip Strauss invented the first successful tire, which was a combination tire and air filled inner tube.
In 1903, P.W. Litchfield of the Goodyear Tire Company patented the first tubeless tire. It wasn't commercially exploited until the 1954 Packard.
In 1904, mountable rims were introduced that allowed drivers to fix their own flats.
In 1908, Frank Seiberling invented grooved tires with improved road traction.
In 1910, B.F. Goodrich Company produced longer life tires by adding carbon to the rubber. Goodrich also invented the first synthetic rubber tires in 1937, made of a patented substance called Chemigum. (info from About.com)