The term autothrottle typically refers to the thrust control of the automated flight management system (FMS) found on most larger commercial aircraft. Simply put, an autothrottle is the autopilot function that increases or decreases engine power. These thrust control systems function by adjusting the fuel flow to the engines in response to a set of desired parameters compared to actual flight input data. These parameters may be set manually by the pilot or extrapolated automatically by the FMS. Autothrottle functions may also be engaged and disengaged at any point by both the pilot and the FMS.
Although far more complex, autothrottle systems may be likened to the cruise control function in an automobile. The system is told how much thrust to apply and does so automatically by adjusting the engine's fuel supply. The source of those instructions is where the story becomes a little complicated. Although the commands to advance or arrest the throttle come from the FMS only, the parameters to issue those commands may come from the FMS or from the pilot. In other words, the FMS may adjust the autothrottle on its own according to prevailing conditions during flight or to parameters requested by the pilot.
Although this probably sounds complex, it really isn't. The FMS constantly monitors the entire flight envelope according to what it has been told about the aircraft's weight, fuel consumption characteristics, exact flight plan or route, amount of fuel on board, and a host of other information. It constantly cross-references this information against real time inputs it is receiving from the aircraft instrumentation regarding present altitude, exact position, airspeed, and rate of climb or descent. It will then instruct the autothrottle to increase or decrease thrust to maintain the desired flight profile. It does this quietly in the background with no input from the pilot who simply monitors the flight progress.
When the flight reaches a point where deviations from the loaded flight plan are required, the pilot can then instruct the FMS to control the thrust according to pilot input. This is typically done by dialing in a desired airspeed or rate of climb or descent on the FMS mode control panel (MCP). In these cases, the FMS is still controlling the thrust but is doing so according to pilot input. Once the aircraft is on the final stage of its approach the pilot will, weather permitting, completely disengage the FMS and autothrottle and fly the last thousand feet or so in by hand.
As mentioned, this device controls the amount of thrust an aircraft's engines develop by increasing or decreasing their fuel flow. In some cases, the system will also physically move the throttle levers in the cockpit as is the case with Boeing and several other aircraft types. In other aircraft such as Airbus products, the throttle levers remain static even if thrust is being manipulated by the autothrottle.