This requires heavy invocation of quantum physics, but I'll try:
The wells are features in the potential energy of a structure. If we were talking about gravitation, the wells would be two trenches in a plane. If you would put a ball on the plane it would move freely, but once it gets into one of the trenches, it would fall down and be stuck there.
But with quantum wells we are (usually) talking about the electromagnetic force acting on a charged particle, e.g. an electron. As long as the electrons stays in the part of the structure where the potential is flat, it can move around freely. But if it gets into the part with the trenches it can emit energy and be stuck in the area were the wells are in the potential.
If the electron would be behaving classically, it would behave as before with the only exception that it cannot leave the well. But if the well is small enough, the fact that the electron behaves as a wave comes into play. The electron does not behave as a ball anymore, but is described by a probability distribution behaving as a standing wave in the well. If you solve the Schrödinger equation for that situation, you see that there is only a finite number of energies the electron can have. If we go back to the analogy of a gravitational potential it would be if a ball could only hover at heights of 1m, 4m and 9m, but not at some height in between (but along the trench it could still move freely).
In a double well, another quantum effect becomes relevant: If the wells are close enough together, the electron can tunnel from one well to the other. That means, the electron can "jump" over the barrier between the wells and can be found in the other well than it was put in initially. This would be like a ball lying in one trench, which is then suddenly found in a parallel trench without being thrown over the barrier separating them.