The iconic old cars in Cuba exist not because Cubans have a thing for vintage American vehicles. Contrary to what people may think, all Cubans would much prefer a new American car, but they are simply not available due to the embargo placed on Cuba in 1962 by President John F. Kennedy after the Cuban Revolution and due to Cuban law.
Prior to the revolution, there was a brisk trade between the United States and Cuba. Because of the lovely Cuban countryside, beautiful beaches, and numerous casinos, the island was a popular vacation destination for Americans. It became even more popular during the Prohibition years when there was a thriving bootlegging (called "rum running") business. The island remained popular during the 40s and 50s, especially with organized crime figures, who used the place as a refuge and money laundering destination.
These people brought their cars with them, and the cars remained long after they left for good, and trade between the two countries dried up. The old cars in Cuba date from the 1930s to 1959. Most are 1940s and 1950s, and these are the large muscle cars that so many Americans covet. They are known as "cacharras" or "bartavias," and you can find these old classics everywhere, from the smallest villages to capital city of Havana. Americans who are lovers of classic cars dream about the day when relations between the two countries normalize and they can freely head to Cuba, buy old classics for pennies and bring them home.
However, the majority of the thousands of cars are rattletraps—and that's being generous. Most owners don't know how to service them, and it's not unusual to find someone who has never changed the oil in the engine. When parts go, there is no replacing them, so a good number of moving parts on any given car will likely be jerry-rigged and makeshift. A car is lucky to have all its door handles and windows, and things may be wired shut, wired together, glued shut, or any number of other oddities. You may have to climb in a window in some cars, and with others you have to hold the door shut while the car is moving.
That said, there is, however, a small fleet of beautifully refurbished old cars in Cuba available for rent to tourists from a government agency. If you're lucky enough to get one of these, it's a wonderful form of independent transportation around the island. You should make arrangements as far in advance as possible as the number available is limited. There is a possibility that the classic cars from America's past will soon become part of Cuba's past.
Cubans are not allowed to buy or sell a car that was not already on the road before the 1959 Cuban Revolution, begun in earnest when the Castro brothers, Ernesto "Che" Guevara, and 79 other revolutionaries sailed the Granma yacht from Mexico to attack Batista's army, so you may be surprised to see modern Peugots from France, Ladas from Russia, and Kias from South Korea on the road. However, these are all government-owned and a normal Cuban cannot buy one.
Raúl Castro recently announced a change that should allow everyday citizens to legally buy and sell cars and houses. Some fear that when the ban is lifted, Cubans will get rid of their old cars to buy new ones, and visitors almost universally hope that the country will realize the cars themselves are a big tourist attraction and make efforts to preserve as many as possible.