MAP = Manifold absolute pressure and is a sensor to measure vacuum, the computer calculates the engine air flow based on the vacuum, throttle position, and intake air temperature. The computer also measures the barometric pressure using the MAP sensor before the engine cranks, this lets the computer compensate for altitude, or how dense the air is.
Typical GM MAP Sensor
MAF = Mass Air Flow and is a sensor that measures the actual air flow going into the engine.
The one above heats a wire and then measures how much the airflow cools it off so it can determine the air flow. While the one below just measures how much the airflow opens the door.
Actual air flow (MAF) is more accurate than calculated air flow. Some cars have both MAP and MAF and use the MAP to fall back on if the MAF fails.
When they fail it causes one or many of the following a very rough running engine, stalls, low power, black smoke as the ECM is delivering fuel based on air flow. They will also set codes related to their systems when there is a problem