Microwave ovens work by electromagnetic radiation. A transmitter inside the microwave (like a mini radio-tower) sets up an electromagnetic field, which sends out microwaves of radiation.
These microwaves hit the water molecules inside your food and make them start wiggling around. This motion creates heat, and voila! Your food gets hot!
Part of the problem with metals is that they are a lot more solid than food. As food gets hot, water just converts into steam and the energy gets released. As the molecules inside a piece of aluminum foil heat up, they don't have anywhere to go. The metal then heats up very rapidly and will eventually catch on fire.
Depending on their shape and surface smoothness, some metal objects will reflect the microwaves instead of absorbing them...which then bounce off the metal in the walls...which then bounce off the object again...and so on until eventually the transmitter itself overheats and becomes damaged.
Another problem is that metals are electrically conductive. If the object has pointy bits, like a fork or the metal filigree on a fancy plate, electrons collect at the edges. When enough electrons build up, an arc can jump between the metal and the electromagnetic transmitter. This is what creates that mini lightening show in your microwave!