Microwave ovens cook food with radio waves, which are just like the waves that carry cell phone conversations, television images, and (of course) radio broadcasts; the only difference is that most of the radio waves in a microwave oven are at higher frequencies than the radio waves used for communication, although many ovens also emit some lower frequency waves that can interfere with cell phones and other devices. In other words, microwaves are simply high frequency radio waves.
Microwaves, like other radio waves, are reflected by metals but pass through many other materials easily. In an oven, microwaves broadcast by a microwave antenna are reflected by the oven's metal walls, which ensures that the microwaves stay inside the oven rather than escaping into your kitchen. As microwaves pass through food some of the microwave energy is converted into heat when the microwaves are absorbed by water and other food molecules.
Because the molecules that absorb microwaves are usually distributed evenly throughout food items, the inside of the food gets as hot as the surface. Although some people say that microwaves cook from the inside out, it is more accurate to say that microwaves cook the outside and the inside at the same time. At least, that's the intent, but many microwave ovens have hotspots where the microwave signal is more intense and can heat food more. Ovens with rotating stands move food around so that the hotspots don't heat one portion of the food too much. In other ovens, the food sits still and a metal reflector (sometimes hidden behind a plastic cover) rotates instead, which moves the hotspots around.
The even heating microwaves provide is not very good for cooking bread. For a crust to form on bread, the outside of a loaf must be heated more than the dough inside the loaf. Conventional ovens, on the other hand, make beautiful crusts and browned foods because they heat food from the outside, and the surface cooks much faster than the inside.