The most simple type of internal combustion engine, it is frequently used in situations where not much power is needed. These typically include small motorcycles, lawnmowers, radio-controlled vehicles etc. It is called a 2-stroke to distinguish it from the 4-stroke engine. While a 2-stroke is actually about 50 per cent more powerful than a 4-stroke of the same displacement, they do not scale well to greater displacements.

It works, as the name suggests, through 2 phases of movements. It's kinda hard to explain without a diagram ... let's see, time for some ASCII ...

      |   |
      |PPP|===== <-- intake pipe
      |PPP|===== --> ejection pipe
      | | |   
      | \ |
      | C\|

S = spark plug
P = piston
C = crankshaft
Above is a simplified diagram of a 2-stroke (there's a complicated tube at the side that ensures that more fuels is burnt, but we'll forget about it for now).

The first stroke, the downward stroke is when the piston is below the ejection pipe. This pushes out burnt gases. It also brings in new fuel.

The second stroke is on the way back up again. As the cylinder goes up, it compresses the mixture of fuel and air coming in from the intake pipe. At the top, the mix is ignited by the spark plug, which pushes the cylinder back down again, simultaneously taking us back to the first stroke and also making the crankshaft turn. The crankshaft turning is the useful side effect we use (usually moderated by a flywheel) to make wheels turn, blades spin, etc.

Update (special thanks to Starrynight): The problem with large 2-stroke engines isn't so much the issue of power, but the issue of emissions. Because of the way they work, the lubricant must be included in the fuel; and when lubricants burn, they create nasty stuff for the environment. Apparently, there are 2-strokes with displacements of more than 1000cc.