What does the term "trimming" most commonly mean in aviation?
In short: trimming neutralizes the force required to keep control surfaces in a specific position. 

Most (if not all) aircraft have some sort of elevator trim control. For example, when the pilot has to keep pulling back on the yoke/stick during a climb, trimming "nose-up" will neutralize that force. The elevator will then remain in the same position without any force required on the part of the pilot. Trimming is usually done by means of a Trim tab on the control surface and is controlled by a trim wheel.

Elevator trimming is used:  


During climbs: to maintain a constant air speed and rate of climb
During descents: to maintain a constant air speed and glide path
During level flight: to maintain altitude and speed


Although elevator trim is the most common, trimming can also be done on rudder (quite common) and ailerons (only on larger aircraft).

Rudder trim is used to maintain coordinated flight without rudder input by the pilot. Many single engine planes with powerful engines require rudder trim to offset the "left-turning tendency" caused by P-factor and propellor wash hitting the rudder. Aircraft without adjustable rudder trim will usually have a fixed trim tab on the rudder.

Most autopilots will control the elevator trim wheel because the servos controlling the elevator could easily be overpowered by the required force.

It's important to note that in most cases (especially on small airplanes) trimming doesn't actually move the control surfaces. It changes the force required to deflect the control surfaces. Although there are many types of trimming devices, the common trim tab makes the airflow do all the work. The trim tab causes the airflow to push the control surface in a specific position. On larger aircraft with hydraulically operated control surfaces, trim tabs are less common (see also this answer to another question)