I was just reading a very cool book called The Fundamentals of Aircraft Combat and Survivability Analysis and Design from my university library. Apparently there are four main ways of directing a missile on a collision course with a target aircraft.
1. Pursuit - The missile tries to turn and point itself at the target for the duration of it's flight. This usually ends up putting the missile behind the aircraft. There are some big problems with this system. First, the missile must execute extremely hard turns in the terminal phase of it's flight when it might be just meters away from the target aircraft. Second, it must be considerably faster than the target aircraft if it is chasing the aircraft. This usually occurs when the missile is cruising and has neither great speed, and little ability to turn.
2. Lead Angle Computation - The missile computes what angle between it's line-of-sight and the aircraft is needed to collide with the aircraft, and for maneuvering targets, this angle is computed and adjusted as fast as possible.
3. Three Point Guidance - The missile's course and maneuvers are based on trying to force it to rest on the line between the target and the launch aircraft. This is computed on board the launch aircraft and relayed by wire or radio. Beam-riding missiles work this way, to the best of my knowledge.
4. Proportional navigation - Based on the fact that if two moving objects maintain a constant line-of-sight angle between them, they are on a collision course. Essentially the missile tries to keep a constant angle between it's longitudinal axis and the line between itself and the target. This is the most common system.
Edited by Genghis