Maximo Gomez Park
This is the place where old Cubans hang out and play chess. It is definitely worth going to watch these guys fight it out. They sip on their insanely strong coffee, puff on seriously long cigars and play some advanced chess or dominoes.
The signs in Spanish will tell you that no spitting or swearing is allowed in the park, so check whether you like the coffee you bought down the street before you walk in.