A torque wrench is a wrench designed to apply a specific amount of force (called "torque") to a nut or bolt. When non-mechanical people hear the word "torque" they often think "isn't that like horsepower?" Torque does in fact refer to force rotating something about an axis (such as the wheels of your car), but that term can refer to anything in that situation; not just how powerful your car's engine is.
Another common example involves the lug nuts that hold the wheels on your car. Lug nuts have a specific amount of toque with which they must be tightened. If too little is used, they might come loose while you're driving which could result in an unstable wheel or the wheel even coming off! If they're too tight you run the risk of something cracking which can also cause the wheels to have problems.
So if these nuts must be applied with a specific amount of force (torque), how does one ensure that exactly the right amount of force is used?
You can make sure the correct amount of force is applied by using a torque wrench!
The most basic style of this type of wrench has a little gauge on it that shows you how much force you are applying. So in this case, if the lug nuts on your cars require 80 pounds of torque, you put the wrench on and tighten it until it says you're applying 80 pounds of torque, and then you're all set.
There are many types of wrenches like this. The basic kind, as mentioned above, is just a wrench with a little gauge on it, usually with a needle indicating the amount of force being used. Some have mechanisms that prevent you from over-tightening past a certain point. These may have a clutch that disengages when a certain amount of force is reached thus preventing any additional force being applied. There are electronic wrenches that have a digital display instead of an analog gauge. There are also hydraulic torque wrenches that are capable of applying a large amount of force through the use of hydraulics. Sometimes this type is used on large pieces of machinery where the more basic wrenches wouldn't work.