Because the two-stroke engine makes power on every down-stroke, they produce roughly twice the power of a four-stroke engine of the same displacement. That is, a 500cc two-stroke will have the same output as a 1000cc four stroke engine - with fewer moving parts (valves, pushrods, rockers, cams), and far less weight.
The simplest two-stroke engines don't even have a dedicated lubrication system. Oil and gasoline are mixed together, and that blend of oil and gasoline lubricates the engine as it passes through, prior to being burned. Although a two-stroke engine is extremely simple and really pretty crude, there is a lot of activity occurring each time the piston travels up or down. Intake, compression, ignition, and exhaust all take place in a single up-down cycle of the piston.
Let's look at a cutaway of a single-cylinder two-stroke engine below.
On the down-stroke (following combustion), at left, the piston is moving downwards. Exhaust gas exits to the top left, as the piston uncovers the exhaust port. The downward travel of the piston also reduces the volume of the crankcase at the bottom, pressurizing it. (The crankcase on a small two-stroke engine is always alternating between positive and negative pressure). Within the crankcase is a combustible fuel/air mixture (with a small amount of oil to lubricate the engine bearings).
As the piston travels further down, it uncovers a transfer port, which allows the air/fuel mixture to move out of the pressurized crankcase and into the combustion chamber at the top. This creates a significant supercharging effect, because of the positive pressure in the crankcase. The fuel/air mixture is actually being forced into the cylinder under pressure. Just like supercharged engines, two-stroke engines have a much lower compression ratios than a four stroke engine that runs at atmospheric pressure. The compression ratio is reduced so that the heat of compression does not prematurely ignite the fuel/air mixture - a symptom known as "pinging" or "knocking". It can ruin an engine.
Back to the previous images though: On the right image, the piston has completed its down-stroke, and is now moving upwards. This increases the volume of the crankcase, which puts it under a vacuum. As the piston moves up, it uncovers the intake, and so the crankcase vacuum sucks in a fresh charge of combustible fuel/air mixture from the carburetor. Also, as the piston travels upward, it closes off the transfer port and exhaust port, allowing the fuel/air mixture to be compressed before ignition.
The above descriptions and diagrams are for a "piston-port" two stroke engine. That is, the up-down motion of the piston covers and uncovers the ports for the carburetor intake, transfer port, and exhaust port.
Hahahaha! After that long explanation, I found a video that does a far better job than I can of explaining how a two-stroke engine works. Meh.
Two stroke engines have a few built-in inefficiencies, not mentioned in the video. You will notice on the above left image that the exhaust port and fuel transfer port are open at the same time. Because of this, some of the fuel/air mixture entering the cylinder is lost out the exhaust before the piston's upward travel finally closes off the exhaust port to begin compression. This doesn't do much for either fuel efficiency or emissions. A lot of engineering goes into minimizing this loss of fuel. Likewise on the right image, as the piston first begins its downward travel, it tends to cause a reverse flow through the carburetor. There are a variety of creative technologies that have been used to minimize both of these issues.
The animated image below shows one method of how the loss of fuel/air mixture out the exhaust is reduced. An expansion chamber is fitted to the exhaust. The expansion chamber reflects the pressure wave of the exhaust pulse back toward the engine, forcing a portion of the lost mixture back into the cylinder. This reverse flow occurs shortly before the exhaust port is closed off by the piston.
Without expansion chambers, two strokes are little better than less fuel efficient versions of four stroke engines. The expansion chamber is what changes a two stroke from a gutless lawn mower engine into a fire-breathing monster.
Expansion chambers typically have a narrow RPM range at which they are most effective, because the precise timing of the return pulse is based on the width and length of the pipe, and therefore will work best at a specific engine speed that is resonant with the expansion chamber dimensions. If you've ever ridden a two stroke motorbike, snowmobile or watercraft, you know what that sweet spot feels like! That resonant point is often called "getting on the pipe" - that RPM range where the engine really pulls hard.
Below, an actual expansion chamber, with silencer beneath.
Another method for reducing loss of fuel from the transfer port out the exhaust is called Schnuerle porting. The transfer port is split into two smaller ports, and the mixture swirls in a controlled way to better sweep exhaust gas out of the combustion chamber, while keeping most of the fuel/air mixture from flowing directly out the exhaust. See image below.
Another problem two-stroke engines have is backflow. As the piston of a two-stroke begins traveling downward, but before the transfer port is uncovered, the piston begins pressurizing the crankcase. This pushes the fuel/air mixture backwards through the carburetor. This backward flow causes two problems: It screws up the fuel/air mixture, because a portion of the fuel/air mixture will pass through the carburetor twice, affecting the ratio of fuel and air. This backflow also reduces crankcase pressure, and therefore reduces the supercharging effect that makes the two-stroke such a powerful engine.
There are a couple of ways to eliminate backflow, and both involve blocking the intake port by mechanical means. The first method someone invented was the Rotary Disc Valve. This is a spinning disk attached to the crankshaft that has a hole cut in it. The hole will only be aligned with the intake when the piston is traveling upward and the crankcase is ready for a new charge of fuel/air mixture.
Another, much simpler, method for solving this reverse-flow problem with two-stroke engines is the use of Reed Valves on the intake. The Reed Valve is a simple one-way valve that prevents the fuel/air mixture from flowing backwards out the carburetor.
The Reed Valve is the little flapper valve on the right side of this animation. The smoke looks about right, considering two-strokes burn the engine lubricant along with the fuel :)
Physically a Reed Valve looks like this. The reed petal is the only moving part. It flexes open and shut as flow changes direction. These need to be replaced at intervals, depending on the material they are made of. Carbon fiber is a preferred material. One of the biggest advantages of reed valves is that they widen the narrow powerband of a two-stroke, so that it feels like it is "on the pipe" (at the resonant point of the expansion chamber) over more of the RPM range.
And... here's where we apply two-stroke power to motorcycles. C'mon, you didn't think I was going to discuss chainsaws and weed whackers did you?
Let's check out a couple of examples of two stroke engine awesomeness. Below is the 499cc Honda V-4 NSR engine. It made over 200 horsepower.
Below, the Yamaha YZR500 engine, another V-4. Imagine how wild they would be if they had modern engine control systems and fuel injection!
Below, the dual-crankshaft Suzuki RG500 square-four engine. Note the Rotary Disc Valve induction system. Photo credit - Phil Aynsley
Needless to say, due to fuel/air flow-through and the need to continuously burn lubricating oil, emission controls were a death sentence for two-stroke street bikes. The last handful were made in the mid 1980s. Two-stroke street bikes are becoming pretty rare. All of the major Japanese manufacturers produced a two-stroke road bike, although Kawasaki appears to have been the most successful, if not the most advanced or road-worthy :)
The air-cooled RD series, a piston-port inline two cylinder air-cooled design (note the cooling fins). Displacement increased over the years from 60, 100, 125, 250, 350 and finally 400cc. The air-cooled RD series ran from 1972 to 1980.
The RZ350. This was a pretty advanced motor. It was an inline two cylinder, with a mechanically adjustable exhaust port height for improved power and emissions (the YPVS logo you see on the engine). The motor had reed valves, and was liquid cooled, with the radiator just behind the front fender. These were built between 1980-1986
People who didn't have to deal with the US Environmental Protection Agency could purchase one of these: An RD500 (or RZ500). This bike housed an actual V-4 liquid-cooled 500cc engine with reed valves and YPVS, just like the layout of a grand prix motorbike. The performance wasn't quite in the same ballpark, but the basics were there, and it probably wouldn't take much to make it run like a GP bike.
Suzuki built the successful GT series, with inline two and three cylinder, piston port air-cooled engines. Engine displacements came in 185, 250, 380 and 550cc. These were not known for being rocket bikes.
Suzuki's GT750 was a very successful two-stroke. It had a liquid-cooled inline three-cylinder engine. This bike was not popular due to its insane power, but due to its comfort, reliability and lack of vibration. This WAS your father's Oldsmobile :), and was nicknamed the "Water Buffalo"
Suzuki also contributed a gem of a bike to the public: The RG500 Gamma:
This bike had a liquid cooled square-four engine, variable exhaust port control, and is the star in one of the videos below.
Soichiro Honda never really liked two-strokes. He spent a great deal of time and his company's money making four-stroke racing motorcycles to compete against his rivals' two-strokes. Eventually Honda moved to two-stroke engines in racing, and then built the street-legal NSR250R for everybody but the US. The NSR250 featured a liquid cooled V-2 engine with adjustable exhaust porting. These were built from 1985 to 1996.
And of course, there was the infamous ill-handling, but powerful Kawasaki 500 and 750.
Below is a vivid example of the power available in a two-stroke a street bike with mid-1980s technology. This guy isn't doing wheelies because he is revving the motor and popping the clutch. He is doing wheelies because the engine pulls THAT hard when it's on the pipe. This isn't a big heavy four-stroke liter-bike. It's a small bike with a crazy-powerful 500cc motor. The best series of wheelies happens at the 3:00 mark. My modern 1000cc sport bike can't do 3rd gear wheelies like that.
Check out these older wheelie-prone 500cc grand prix bikes... the unrefined peaky power spun the rear tire up too easily, causing a lot of rear wheel slip-and-grip crashes.
Amazing how much power a small two-stroke engine can make. 100 mph and still loft the front wheel way in the air? No problem! These guys were trying to apply as much power as possible without flipping the bike over. Later it was learned that bikes accelerate faster when the front wheel stays on or near the ground.
How about drag racing? 1970's technology two-stroke bikes with air-cooled 750cc motors vs. modern liquid-cooled, computer-controlled four stroke motors with twice their displacement!