According to dictionary.com:
A mechanical or hydraulic device for changing the ratio of torque to speed between the input and output shafts of a mechanism.
This is the part of an automatic transmission that connects the engine flywheel to the automatic gearbox, and is filled with oil or transmission fluid. It contains three parts - an impeller, which is driven by the engine itself; a turbine, which is connected to the automatic gearbox; and something called a reactor, which looks like a gear, but has vanes instead of teeth. (Also known as a stator.)

The engine flywheel turns the impeller, which starts to move the fluid inside the housing. The fluid is channeled past the vanes of the reactor, which sits inside the diameter of the impeller itself, and then further inward to the turbine. In this way, the engine's power is applied to the turbine (and thus the input shaft to the gearbox) in a smooth manner, as it takes some time for the turbine to start spinning at the same center speed as the impeller.

The faster the impeller is moving, the more fluid it will pump into the housing, and at higher pressure. This helps to maintain the power curve, since the fluid is really the thing that does the work.

Torque converters are set to stall at a certain speed, and can also be made to stall by mechanical means. In either case, the hydraulic pressure drops off suddenly, which has the effect of de-coupling (well, not physically - the coupling is fluidic in nature) the impeller from the turbine. This stops power transfer between the engine and the gearbox. Automatic transmissions typically last much longer than manual transmissions because of this - the gearbox is essentially disconnected from the engine during shifting, so the gears are not made to slam into each other at the instant of engagement. It is also why an automatic transmission makes little or no noise during shifting. (Sports automatic transmissions usually make a satisfying "clunk" sound.)

Torque converters are why you don't have to select neutral gear when you are coming to a stop. The converter is engineered so that when the engine is at idle speed, the hydraulic pressure inside the housing is low. Low pressure equates to less torque transferred, so the push on your wheels is very light.

The auto enthusiast's main complaint against torque converters is that you lose some power in the converter, typically up to 10 or 20 horsepower. There are various tricks to decrease the losses, however. The most obvious method is to design a more efficient torque convertor, but that happens anyway as the years go by. Another trick is to have the ECU monitor pressure inside the housing, and to "flood" it to a certain pressure according to the speed at which one of the shafts (input or output) is turning. This has the effect of creating a computer-calculated optimum level of pressure within the housing, thereby improving coupling between the impeller and the turbine. Finally, a lockup clutch can be used to engage the turbine and the impeller directly, rather than using fluid; this transfers 100% of the power. (It is not appropriate to do this all the time, as you cannot multiply torque when these two parts are direclty engaged. Read on.)

One of the coolest things a torque convertor can do is to actually multiply torque coming from the engine by two or three times. Mechanical engineers have known for centuries that you can trade speed for torque. For example, as rotational force is transmitted through a reduction gear, the speed drops, but the torque increases. This is useful for situations in which power is more important than maintenance of speed. Freeway onramps are a prime example of such an occasion.