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Autopilots are primarily mechanical or electronic systems. Some autopilots are composed of pure luck; these are capable of flying for several hours while turned off.
Autopilots can maintain "straight and level flight". This contrasts to the wobbly and uneven flight followed by a drunk pilot. In this mode an autopilot can keep the passengers calm without raising their suspicions.
More advanced autopilots can also perform Automatic landing. In the early days this consisted of fog and a big mountain. Nowadays it is a more sophisticated system of radar, gyroscopes, computers, fog, and a big mountain. There is also an element of surprise.
Most autopilots operate with inertial guidance systems. Such systems are very difficult to get going but once moving they work resonably well, at least in a straight line. A major problem with such guidance systems is accumulated error, causing them to get their directions wrong. Often the first sign of this is when the guidance system says "On your left you can see the airport", when in fact it's the coastline, and you realise that you need to turn off the automatic landing. A common cause of error is gyroscopic drift, an effect of the Coriolis force. Because the Earth is spinning the gyroscope appears to rotate as quickly as the Earth: once per day. Hence it falls over and the compass gets stuck. To counteract this, the gyroscope is forced to spin extremely quickly, but a side-effect of this is that it gets very dizzy. Hence it falls over and the compass gets stuck. A popular fix for this is to do a barrel-roll, or preferably make the aeroplane do a barrel-roll, which "topples" the gyroscope, along with any passengers.
The concept of an autopilot has been enhanced in recent years to provide complete computer control over the entire aircraft, supposedly in an attempt to stop the toppling of passengers after the tippling of pilots. Such systems are completely infallible and so it is not until they go wrong that people realise that a manual override would have been a useful addition.
edit Autopilot in places that you would not expect
Interestingly, autopilots occur in nature. Some birds are able to sleep while flying. A similar feat is apparent in humans who frequently go into "autopilot" during marital sex, while driving on motorways or listening to a politician.
An automatic landing feature is also inherent in all animals that fall off mountains, particularly in the fog. Usually this a single-shot feature and not easily repeatable, but it rarely fails.