You think of a normal well as a hole in the ground. A quantum well is analagous and represents the same concept: Energy. For example, if the hole in the ground was 10 meters down, then you would need an energy of your mass times the height of the well * the gravitational acceleration constant (~10 meters per second squared) to rise vertically out of the well.
Now lets apply this concept to the world of quantum mechanics. Lets use an electron as the item trapped in a well (which is typically done). Lets look at the simplest case of electron motion: 1 dimension, or a straight line. Lets say there are atomic forces on either side of the electron pushing against it, preventing it from going past them. They have to push with a certain energy and that is the energy that the electron must overcome to move past them (since they are blocking its path). Well, now introduce the famous schroedinger equation that defines electron motion and introduces the weird world of quantum mechanics. Solving that equation, knowing the energy with which the atoms on either side 'push' on the electron, produces a startling result: the amount of energy the electron has is quantized! for example the electron can have 1 joule or 2 joule, but nothing in between! The energy levels of the electron can ONLY exist in these discrete values until it has enough energy to overcome the barrier and 'escape'.
ANother interesting result of the quantum well, is that the electron motion is stationary!!! Electrons do not 'bounce' back and forth in a well like you would imagine a rubber ball hitting a wall. The schroedinger's equation describes a wave like nature of an electron and therefore there will be certain 'vibrations' of this wave that don't look like they are moving at all. You can verify this by vibrating a string faster and faster until you don't see the string moving. Increasing it faster and faster makes it move again until it comes to the next vibration that looks standing still (this is called a standing wave and only occurs at certain frequencies or 'vibrations').
These quantum weirdnesses (discrete energies and wave like motion of matter) result in strange behaviour. For example, there is a small chance for the electron to 'tunnel' out of the well, even if it doesn't have enough energy. Kinda like you all of a sudden appearing out of the hole in the ground. Well, it wouldn't happen in the 'real' world, but when you are as tiny as an electron, this is an everyday thing!
You can take advantage of this 'weirdness'. A practical example is in your CD player. The laser that reads information of your discs is a quantum well laser and confines electrons by sandwiching materials together: simplistically, like a thin piece of capicolla between thick italian bread (forming the quantum well). The electrons are confined to the thin piece of capicolla, and without going into a lot of materials science, produce a really efficient laser! Keeping the middle layer thin is important to confining the electrons. If the material were relatively 'thick' then why would the electrons care about the boundaries and be confined when they can 'roam' free everywhere else in the material! We are talking nanometers here (billionths of a meter).
Quantum weirdness is a reality and verified to unprecedented accuracies and is common place in our technological world.
Paul Speziale, B.S., Eng Physics Grad Student, McMaster University, Ontario
'Our job in physics is to see things simply, to understand a great many complicated phenomena, in terms of a few simple principles.'