ECU - electronic / engine control unit of the engine. controls some of the cars elctrical systems. It's main use is to control the ignition systems in the engine. Namely spark timing, fuel injections, and valve timing. Changing or modifying some of the ECU's stock programming would allow greater fuel efficiency, improvement on engine power, and even lower pollution levels.
ECU mapping involves modifying the presets programmed on the ECU to achieve the desired effect. Some vehicles needs to be mapped manually in order to improve performance, while other modern cars have ECU's that simply have to be updated to improve performance.
ECU tuning and mapping required great technical knowledge. So in my opinion if you're interested in getting one, consulting a professional would be the best. Because although ECU tuning could greatly increase the car's performance, wrong ECU tuning or mapping could also greatly decrease the car's power or worse.