What kinds of cars did people drive back in the 1960s?

Mostly the same brands, even some of the same model names as today, since they were first marketed in the 1950s and 1960s.

There were Ford, Lincoln, and Mercury models: Ford Fairlane, Galaxie, and Falcon, and Mercury Comet. The 1960s saw the first Mustangs, and improved models of the Thunderbird, which changed from sports car to luxury car in 1967. 1960 was the last year for the unappreciated Edsel (1958-1960).

GMC trucks were around, along with Chevy trucks, and passenger cars like the Chevrolet Impala, Bel Air, the air-cooled Corvair, and racy Corvette. Buick had the Wildcat and the Riuviera, and the compact Skylark. Oldsmobile had the Cutlass, F-85 and Vista Cruiser. Cadillac produced various models of its Calais, DeVille, and Fleetwood.

Plymouth had the Fury, and the Belvedere became the Plymouth Satellite - Dodge also built the Dart. The Dodge Challenger did not appear until late 1969.

Volkswagens were introduced in new styles other than the Beetle: the type 3 featured a Fastback, Notchback, and Squareback, and the Karmann Ghia appeared in 1961. The now-classic VW bus was produced as well as the wider Van through the 1950s and 60s.

There were models of some earlier car companies: AMCproduced the Rambler American, the Rambler Classic wagon, the Ambassador, the AMX, and the frog-tailed Marlin. The Studebaker line rolled out the Avanti, but ended its run with the Cruiser in 1967.

The 1960s saw the beginning of Asian imports: the Toyota Crown, early Toyota Coronas, the Subaru 360 city car, and the earliest Datsun "Nissans" such as the Laurel.