The Two-Stroke Cycle
The following animation shows a two-stroke engine in action. 

This figure shows a typical cross flow design. In this figure you can see that two-stroke engines are ingenious little devices that overlap operations in order to reduce the part count to a minimum.

You can understand a two-stroke engine by watching each part of the cycle. Start with the point where the spark plug fires. Fuel and air in the cylinder have been compressed and when the spark plug fires the mixture ignites. The resulting explosion drives the piston to the right. Note that as the piston moves to the right, it is compressing the air/fuel mixture in the crankcase. As the piston approaches the bottom of its stroke, the exhaust port is uncovered. The pressure in the cylinder drives most of the exhaust gases out of cylinder, as shown here:

As the piston finally bottoms out, the intake port is uncovered. The piston's movement has pressurized the mixture in the crankcase, so it rushes into the cylinder, displacing the remaining exhaust gases and filling the cylinder with a fresh charge of fuel, as shown here:

Note that in many two-stroke engines that use a cross-flow design, the piston is shaped so that the incoming fuel mixture doesn't simply flow right over the top of the piston and out the exhaust port.

Now the momentum in the crankshaft starts driving the piston back toward the spark plug for the compression stroke. As the air/fuel mixture in the piston is compressed, notice that a vacuum is created in the crankcase. This vacuum opens the reed valve and sucks air/fuel/oil in from the carburetor.

Once the piston makes it to the end of the compression stroke, the spark plug fires again to repeat the cycle. It's called a two-stoke engine because there is a compression stroke and then a combustion stroke. In a four-stroke engine there are separate intake, compression, combustion and exhaust strokes.

You can see that the piston is really doing three different things in a two-stroke engine:

1. On one side of the piston is the combustion chamber. The piston is compressing the air/fuel mixture and capturing the energy released by ignition of the fuel.
2. On the other side of the piston is the crankcase, where the piston is creating a vacuum to suck in air/fuel from the carburetor through the reed valve and then pressurizing the crankcase so that air/fuel is forced into the combustion chamber.
3. Meantime, the sides of the piston are acting like the valves, covering and uncovering the intake and exhaust ports drilled into the side of the cylinder wall.

If you have ever used a two-stroke engine, you know that you have to mix special two-stroke oil in with the gasoline. Now that you understand the two-stroke cycle you can see why. In a four-stroke engine, the crankcase is completely separate from the combustion chamber. In a four-stroke engine, therefore, you can fill the crankcase with heavy oil to lubricate the crankshaft bearings, the bearings on either end of the piston's connecting rod and the cylinder wall. In a two-stroke engine, on the other hand, the crankcase is serving as a pressurization chamber to force air/fuel into the cylinder. Therefore the crankcase cannot hold a thick oil. Instead, the oil you mix in with the gas is how the crankshaft, connecting rod and cylinder walls are lubricated. If you forget to mix in the oil, the engine isn't going to last very long!

Disadvantages of the Two-Stroke Engine
You can now see that two-stroke engines have two important advantages over four-stroke engines: they are simpler and lighter, and they produce about twice as much power. So why do all cars and trucks use four-stroke engines? There are four reasons:

1. Two stroke engines don't last nearly as long as four-stroke engines. The lack of a dedicated lubrication system means that two-stroke engine parts wear a lot faster.
2. Two-stroke oil is expensive and you need about 4 ounces of it per gallon of gas. You would burn about a gallon of oil every thousand miles if you used a two-stroke engine in a car.
3. Two stroke engines do not use fuel efficiently, so you would get lower MPG numbers.
4. Two-stroke engines produce a lot of pollution. So much, in fact, that it is likely that you won't see them around too much longer. The pollution comes from two sources. The first is the combustion of the oil. The oil makes all two-stroke engines smoky to some extent, and a badly worn two-stroke engine can emit huge clouds of oily smoke. The second reason is less obvious but can be seen in the following figure:
   

   Each time a new charge of air/fuel is loaded into the combustion chamber, part of it leaks out through the exhaust port. That's why you see a sheen of oil around any two-stroke boat motor. The leaking hydrocarbons from the fresh fuel combined with the leaking oil is a real mess for the environment.