An elevator is a control surface of an aircraft that controls pitch. Moving elevators up or down pushes the tail of the airplane to go up or down, which is one of the ways that an airplane can change its altitude. Elevators are controlled by the control stick, and can be moved by cables or a fly-by-wire system.
When the flying pioneers were inventing gliders and other primitive flying machines, they though that making machines that looked and functioned similarly to birds would be the key to flight. Of course, birds have tails, which are like horizontal stabilizers, but they do not have features comparable elevators. That is to say, the elevator is not something found in nature, but a movable tail (in the case of the pioneers, a movable horizontal stabilizer) is. Being inspired by birds is the reason that the inventions of Chanute, Lilienthal, and even the Wright brothers built their flying machines with movable horizontal stabilizers (and sometimes an entirely mobile tailplane).
Although they come in diverse shapes and styles, the elevator is a feature that all airplanes have in common. The reason for this is because the elevator is essential in nearly all phases of an airplane's motion, including flight, takeoff, and landing. Elevators are usually located on the trailing edge of the horizontal stabilizer (also called tailplane) of an airplane. They may be combined with the rudder(s) in a v-tail, which is called a ruddervator, as is the case with the Beech Bonanza. They may also be combined with the ailerons, which is called an elevon, like in the Aerospatiale Concorde. Some elevators comprise the entire horizontal stabilizer.
Because the elevators affect the pitch, having the control stick in the neutral position may still result in the airplane going up or down. One of the factors in this happening is that the function of the horizontal stabilizer is effected by airspeed. For example, at airspeed of zero, the motion of the horizontal stabilizer has a negligible affect on the overall motion of the airplane, but when taking off, the horizontal stabilizer provides enough down-force to push the tail of an airplane down, causing the nose to rise. In order to keep the airplane at the desired pitch while still having the control stick in the neutral position, trim tabs are needed.