Unlike a traditional oven, which first heats the air which in turn heats the food, a microwave oven cooks directly by bombarding the food with microwave radiation. This is why when you turn on the oven, and then open it up it will release a blast of heat in your face. Then you turn on the microwave for 2 days, open it up, and no heat at all. The concept behind this is friction, which I will comment on later.
But first, I am sure you don't know what a microwave is, well here is a good explanation: Microwaves are a type of electromagnetic radiation just like radio waves, X-rays, and visible light. The only thing that makes microwaves different is the frequency of the waves that make up the radiation. The microwaves which are used to cook food have a higher frequency than that of radio and TV waves but have a lower frequency than those of visible light and X-rays, which is why you can't actually see them. In fact, microwaves are more like high frequency radio waves than anything else. Unlike radio waves, which travel in every direction, microwave radiation is highly directional.
As to actually cooking something with microwaves, the process is quite simple. Once the microwaves are generated(taking a few seconds), they are directed at the food. The microwaves enter the food and agitate its molecules. The rubbing of the agitated molecules causes friction, and it is this friction which heats the food. The frequency of the microwaves (approximately 3 giga Hertz) was chosen because it is especially effective at agitating water molecules. Since food is mostly water, the microwaves are easily absorbed and heat the food quickly. But the microwaves do not go through certain products such as tin foil. This is why when you wrap a frozen baked potato in tin foil it will not cook or warm up.
The oven itself is based on simple technology and actually has been around for several decades. It was difficult to convince people that the microwave radiation used to cook the food is unlike the radiation released by X-ray machines and nuclear bombs. Instead, it is more akin to the radiation emitted by heat-lamps and radio stations. It was the public's misconceptions that were difficult to overcome - not the technology.
The most mysterious part of the microwave is in the generation of the microwaves themselves. Most ovens generate microwaves with a device called a magnetron. A magnetron is composed of a thick outer cylinder with a thin tube inside it. It works on a principle similar to the tubes used in televisions and computer monitors. Whereas the tubes in televisions cause electrons to jump from a plate in the back and fly toward the screen at the front, a magnetron causes electrons to jump from the outside tube and fly to the inside tube. As they fly inward, a magnetic field makes the electrons circle around the inside of the cylinder. Weird things begin to happen when electrons move in a circle. The rotation of electrons is what causes motors to work and cellular phones to transmit. It is also what causes the magnetron produce microwaves.
Once the microwaves are produced, they are emitted upward through the top of the magnetron and into a hollow, rectangular, metal conduit called a wave guide. The wave guide is curved so that it bends the microwaves horizontally along the top of the oven. While in the top of the oven, a small metal fan, called a stirrer, scatters the microwaves down into the oven chamber where they are absorbed by the food. The absorbed microwaves cook the food, accomplishing our goal.
So, the next time you use your microwave to burn popcorn, remember that radiation is heating your food.