Torque is a twisting or turning force that tends to cause rotation around an axis, which might be a center of mass or a fixed point.
Torque can also be thought of as the ability of something that is rotating, such as a gear or a shaft, to overcome turning resistance. It is generally measured in pounds per foot or newtons per meter.
Torque is sometimes confused with work, which is defined as force applied over a distance. However, torsion (twisting) is integral to the definition of torque. Both torque and work are mechanical manifestations of energy.
Torque is commonly spoken of in relationship to automobiles; it's the force that makes cars capable of smoking brake burns and fast acceleration from low speed. Increased torque can be attained by gearing or enhanced engine performance (through modification). Generally, higher displacement engines produce more torque. Diesel engines are known for high torque and low horse power, which makes diesel engines suitable for hauling loads. When transporting heavy loads, starting fast from a stop, or moving gears under resistance torque is a more important force than horse power.
Khan Academy provides an introduction to torque: