The Peltier Effect takes place when an electrical current is sent through two dissimilar materials that have been connected to one another at two junctions. One junction between the two materials is made to become warm while the other becomes cool, in what amounts to an electrically-driven transfer of heat from one side of the device to the other. This transfer can be so dramatic as to bring the cool side below freezing. That they be different is important because they need to have different electron densities in order for the effect to work. When they are connected to each other by an electrical conductor, a new equilibrium of free electrons will be established.
Potential migration creates an electrical field across each of the connections.
When current is subsequently forced through the unit, the attempt to maintain the new equilibrium causes the electrons at one connection to absorb energy, those at the other connection to release energy. The severity of this effect not surprisingly increases with the amount of current sent through the unit. This should be your first clue that Peltiers are serious electricity hogs. More on that later. It's worth noting that the abstract unit depicted above is both simplified and not quite complete in its description of an actual retail Peltier device.