The concept that is at the core of any heat pump is 'selection for higher energy particles' at the atomic level.
Heat is simply motion; random motion of particles bouncing against each other, vibrating in place when held by atomic forces, etc. Just atoms, molecules, electrons, etc. bouncing around. Because this _random_ motion, some of the particles will have more kinetic energy than others.
If you could somehow _select_ for the highest energy particles, then the average energy of the particles that remain will be reduced, and the temperature of the particles that remain will go down.
Luckily this segegation of energy levels happens all over the place. For example, when a liquid boils, it is the molecules with the highest energy that escape first. If you were to take a flask of water and pull a sufficient vacuum on it, then the water would boil at room temperature, and get colder.
The transition from liquid to gas represents an 'energy barrier', and only the highest energy particles have sufficient energy to get over this barrier. When particles climb this barrier the heat contained in the remaining material goes down. Similarly, when particles fall down this barrier and condense into the liquid, the heat contained in the fluid increases.
Now, if you simply have a closed container, after some of the liquid evaporates, the volume of the container will become saturated with the vapor, and you will see condensation of vapor back into fluid. The net result is equilibrium, with the same amount of energy being deposited through condensation as is absorbed through evaporation. But by adding a pump to remove the vapor, you can make the cooling process continue.
In an ordinary freon cycle heat pump, you have a region in which freon evaporates, absorbing heat. You then use a pump to remove the vapor and concentrate it elsewhere. The freon condenses on the high pressure side of the pump, adding heat to the fluid on the high pressure side. By selecting pump pressures correctly, you can absorb heat at the desired cold side temperature, and then reject this heat at the desired hot side temperature.
So we see the four pieces of the system: particles climb an energy barrier, selecting for the particles with the highest kinetic energy. The kinetic energy of these particles is converted to potential energy. Then some external energy input is used to move these particles. Then the particles fall down the energy barrier _in a different location_, converting their potential energy back into kinetic energy, and releasing heat. The particles are allowed to cool down to ambient temperature and returned to the evaporating side of things.
A peltier effect device follows this exact same general principal. The particles are no longer freon molecules, but instead individual electrons. In conductors, the random vibrations of electrons are one of the things that carry the heat of the object. By selecting off the highest energy electrons you can lower the temperature of the object.
At any junction of dis-similar conductors, there is a bit of potential energy barrier; electrons going in one direction will have to have slightly higher potential energy, and the electrons that climb this potential energy barrier are the highest energy electrons.
Of course, if you simply have a junction sitting there, as many electrons will fall down the potential wall as climb up it: equilibrium. But apply a voltage, and electrons that happen to make it up the barrier will get tugged away from the junction. Net result: heat absorbed at the junction. Since electrons must flow in a closed circuit, the implication is that there must be at least one other junction someplace else, and that heat will be rejected at that junction.
It gets more complicated when you try to explain why there is a potential energy barrier between conductors, and what happens with 'P-type' semiconductors. I'm still pretty fuzzy on these latter bits. But the essential part is the selection of electrons for highest kinetic energy, and the conversion of that kinetic energy to potential energy as the electron moves through the junction.