We use them almost daily and yet many people have no idea how a microwave oven works.  Here are the basics courtesy of the folks at HowStuffWorks.com:

A microwave oven uses microwaves to heat food. Microwaves are radio waves. In the case of microwave ovens, the commonly used radio wave frequency is roughly 2,500 megahertz (2.5 gigahertz). Radio waves in this frequency range have an interesting property: they are absorbed by water, fats and sugars. When they are absorbed they are converted directly into atomic motion — heat. Microwaves in this frequency range have another interesting property: they are not absorbed by most plastics, glass or ceramics. Metal reflects microwaves, which is why metal pans do not work well in a microwave oven.
You often hear that microwave ovens cook food “From the inside out.” What does that mean? Here’s an explanation to help make sense of microwave cooking.
Let’s say you want to bake a cake in a conventional oven. Normally you would bake a cake at 350 degrees F or so, but let’s say you accidentally set the oven at 600 degrees instead of 350. What is going to happen is that the outside of the cake will burn before the inside even gets warm. In a conventional oven, the heat has to migrate (by conduction) from the outside of the food toward the middle (See How a Thermos Works for a good explanation of conduction and other heat transfer processes). You also have dry, hot air on the outside of the food evaporating moisture. So the outside can be crispy and brown (for example, bread forms a crust) while the inside is moist.

In microwave cooking, the radio waves penetrate the food and excite water and fat molecules pretty much evenly throughout the food. No heat has to migrate toward the interior by conduction. There is heat everywhere all at once because the molecules are all excited together. There are limits, of course. Radio waves penetrate unevenly in thick pieces of food (they don’t make it all the way to the middle), and there are also “hot spots” caused by wave interference, but you get the idea. The whole heating process is different because you are “exciting atoms” rather than “conducting heat.” 

In a microwave oven, the air in the oven is at room temperature, so there is no way to form a crust. That is why microwavable pastries sometimes come with a little sleeve made out of foil and cardboard. You put the food in the sleeve and then microwave it. The sleeve reacts to microwave energy by becoming very hot. This exterior heat lets the crust become crispy as it would in a conventional oven.