As a result of the US economic blockade on Cuba, there aren't very many new cars in Cuba. However, Cuba has probably the world's largest supply of old American cars. There are between 50,000 and 100,000 pre-1960 US cars being used in Cuba today.
These cars are still heavily in use every day and are kept up well. Many parts have fallen off, and Cuba cannot purchase spares from the US. Sheet metal is commonly found holding these old cars together.
Another popular form of transportation is the bus system. You will see large numbers of people waiting at bus stops all over the cities. Cuba has normal buses as well as Camel buses, which hold hundreds of people, more often than not carrying many more than they're designed for.
Many people walk when possible. In Havana and other large cities, places of work are often too far from housing to make this possible, however.
What automobiles Cubans have, they share. In some small towns, state-owned cars have to stop at the end of town and check with an official. The official finds out where they're going so that people in the town that need a ride there can go along. Many people climb into the backs of trucks. Families driving down the street commonly pick up people and give them a ride. This organized system of hitchhiking allows Cuba to maximize the usefulness of its aging automobile fleet.
The last, and most popular, form of transportation is the bicycle. Cuba has and will buy 1.2 million bikes from China. It is also developing its own bikes (the Chinese bikes were unpopular) and some people are quite proud of them. The government makes bikes available to citizens for just $6. About 30% of Havana's bicycles are used every day.
There is also an airline system in Cuba which links all of the major cities. Obviously in a third world country, few Cubans travel this way.
Lastly, there are several railroads connecting Cuba's cities. This is a good way to get around the country.