In axis and allies 1941 do fighters have to take off to participate in an attack?
The basic principle is that an air unit ends its turn when it "lands" on a space or a carrier. So if a fighter flies two spaces, fights, and lands on a carrier with its third space, its turn ends, and it cannot fly to an adjacent fourth space if the carrier is attacked. (The air unit is refueling while this happens.)

However, an air unit on a carrier can defend its carrier, which is to say that a player can choose to take fighters as casualties before, or instead of the carrier. (If the carrier is UK, and a fighter is US, the UK player needs the consent of the US player.) But most players will sacrifice a fighter to allow the carrier to be taken as a casualty last, because if it is sunk, the fighters are lost anyway.