A microwave works with a magnetron, a vacuum tube that generates small waves (microwaves) of energy. These waves go straight into the food and heat the molecules in it directly from inside. For a long time it was thought that this process, of cooking the food from the inside rather than the outside, would destroy the nutrients in the food, but it turns out that in fact, the shorter cooking time in the microwave helps to preserve more nutrients than slower cooking.

In a conventional or convection oven, the food is heated by an element (source of heat) which warms up the air in the oven. This gradually heats the food from the outside. With a fan oven, the process is the same but the fan causes the hot air to move quickly around the oven and this speeds up the cooking.

You can see exactly how a microwave works here.