As you probably know if you are reading this article, a torque wrench is a type of wrench designed to apply torque (force around an axis) to a bolt or nut. Torque wrenches are distinguished from other types of wrenches because they are able to measure and/or control the amount of force that is applied allowing the proper amount to be used and ensuring that the bolt or nut is neither too loose nor too tight.
Let's start with the most simple designs and work out way up.
A simple torque wrench is usually just a long wrench with an analog needle on it. When the wrench tightens a bolt, the needle will move and point to the value representing the amount of force being used (for example, 50 pounds of force). These will occasionally need to be calibrated, and since the needle is often exposed it may be bent or moved accidentally and no longer point to the correct value.
There are also electronic torque wrenches that work basically the same, except instead of an analog gauge they have a digital display. The advantage to this is that it can be easier to read because the value is shown in numbers as well as being easier to read with less light (assuming it is bright or uses a backlit display) such as if you're working behind a big piece of machinery that is blocking the light.
There are torque wrenches that can be set to a certain level of force that will stop applying force once that level is reached to help prevent over-tightening. With this type you don't have to focus as much on the gauge because it will stop tightening at the correct force.
Hydraulic torque wrenches use hydraulics to tighten or loosen a nut or bolt. This type of wrench uses a predetermined amount of torque and can operate more quietly than similar pneumatic wrenches, not to mention many people feel hydraulic wrenches are a better choice because of the precise nature involved in tightening bolts.