A regeneratively cooled rocket is a rocket whose engine uses (usually cryogenic) rocket fuel to keep the combustion chamber, throat or nozzle from melting. It does this by passing the fuel through pipes around the hot metal bits.
The fuel is heated somewhat by this, and then gets passed into the combustion chamber where it is mixed and is combusted. The preheating actually means that better performance is achieved because the propellents need less energy to reach the combustion temperature; however the rocket engine is more complex to build, heavier and more expensive. Still, for large rocket engines this is one of the most popular techniques, for example the Space Shuttle uses it on the main engines (SSME).