Eeezee's answer got it almost all right. I don't design microwave ovens. But 40 years ago I did research in Physical Chemistry using microwaves to measure molecular properties, so I know the underlying reason that the ovens work.
Molecules naturally rotate in space. Moreover, some parts of molecules also rotate with respect to the rest of the moleulce, and this usually is faster because the parts are smaller than the whole molecule. Now, moleulces are composed of many atoms linked together by sharing electrons in bonds, but the sharing is not always equal - some atoms pull electrons to themselves harder than others. The result is that in many molecules there is an uneven distribution of electons, and the moleule has an "electric dipole". That is, one end tends to be more positive, and the other end more negative. This is NOT the same as ions. Now we put those dipolar molecues into a rotating electric field, and the dipoles will try to line themselves up with the field. Well, more precisely, they are tumbling around anyway. But if the frequency of the applied field is very similar to the rotational frequency of the molecule anyway, the field will temporarily couple with the molcule to make it align a little better. In doing so the field does work on the molecule (by applying a torque force over a non-zero distance), thus increasing the molecule's rotational energy. Now, both rotational and linear vibrational energy are the forms we experience as the heat in an object, so increasing rotational energy means it gets hotter.
It happens that the natural rotational frequency of many small- to medium-sized organic molcules in the liquid state is between 1 and 10 GHz. That is called the microwave region because the wavelengths are 30 to 3 cm. For smaller parts (substituent groups in organic chemical parlance) the frequency may be up into the 50 GHz region (6 mm wavelength). For liquid water it's from 0.5 to 2 GHz. So if you apply an electric field at microwave frequencies to water and food, the molecules will absorb energy from the applied field and get hot.
Let me add a few things to others' posts.
6. Not in a microwave oven - mostly metals for two reasons. One is that they will get hot from the electric currents induced in them. But that can actually be used - there's a thin film of metal in the bottom of microwave popcorn bags. But the other problem is that they generate reflected microwave signals in the oven, and those reflections can upset the operation of the magnetron tube with the possibility of damaging it.
7. Hotspots also can result from different food components. For example, fats are very good at absorbing microwaves and getting hot. But they do not boil off as steam when they reach 100 C, so they can actually get much hotter than that.
9. Efficiency. Microwave absorption by food is complete - 100% of the microwave energy output from the magnetron is absorbed by the food. BUT the magnetron tube itself is less than 50% efficient in converting household electricity to microwaves. So compared to a conventional oven (which converts 100% of its electricity supply to infrared heat), the microwave oven is is less efficient when operating, but it may operate over a much shorter time. You may use it for 2 minutes to heat up a burger, but you'd never heat that burger in a regular oven in only 4 minutes. On the other hand, if you're cooking a roast for an hour in a microwave vs. 3 hours in a regular oven (regular ovens do not run their heating elements all the time), the microwave oven uses up more electrical energy.