A torque wrench is a tool used to measure the amount of torque applied to a fastener (typically a nut, bolt, or screw) when tightening it. Torque is defined as the amount of force applied, multiplied by the length of the lever arm used to apply it. In layman's terms, torque is how tight the fastener is.
In the United States, torque is measured in foot-pounds. In the rest of the world, torque is measured in Newton-Meters.
Different torque wrenches have different ways of displaying/communicating the amount of torque applied. Some have an analog gauge with a needle. Others have digital readouts. Some torque wrenches click for each additional foot-pound of torque applied.
Many moped, automotive, and motorcycle manuals list specifications for the amount of torque to apply to different parts when tightening them. Lubricant is sometimes applied to a fastener being tightened, and this drastically affects torque readings. Typically, adding lubricant makes a fastener tighten more than it would without lubricant. For the most part (and this is open to dispute), moped manuals assume un-lubricated (or 'dry') tightening.
Torque specifications are useful, because they let the technician know what an appropriate level of tightness is. If you over tighten a fastener, you run the risk of stripping threads. If you under tighten, you risk the fastener vibrating loose, or not holding together the parts it was designed to hold together.