A Fighter is a defensive ship that can be used to knock out incoming attacks. It works very much like a Drone, in that it sits in your SpacePort until it detects an incoming attack, and then it launches and destroys the attacker. However, it's better than a Drone in that it usually is not destroyed in the battle. See ship to ship combat for details.
You can build a Fighter after you've built a Pilot Training Facility.
After being used in combat a fighter goes into "reset". If the fighter is defending the body where its controlling spaceport is located, the reset time is based on traveling 2 map units (flying into space and back). If the fighter is defending a remote body as reinforcements away from its controlling spaceport then it is sent home as its reset action.