A quantum well is a device for storing electrons or other charged particles. They are used in many different types of modern technology, including computers, lasers and solar panels.
Imagine a ball rolling into the Grand Canyon, it will fall to the bottom and cannot get back out unless a lot of energy is given to it (for example, someone at the bottom of the well could fire it out of a cannon!). A quantum well works in a very similar way. Electrical forces keep charged particles (such as electrons) inside the well, just like gravity keeps a ball inside the Grand Canyon.
A second important difference is that quantum wells are very small (about 1000th of the width of a human hair). At these scales, quantum effects come into play. This means that the electron may only have certain, fixed amounts of electrical energy. Think of this as being like standing on a ladder - you can only stand on the rungs, not the gaps in-between.
When the electron moves up its ladder of permitted energies, it must absorb exactly the right amount of energy to do so. This is like putting enough effort into raising your leg to reach the next rung of the ladder without overstepping it. Similarly, when the electron moves down its ladder of permitted energies, it gives out a burst of energy of exactly the right size.
Packets of light
Einstein won a Nobel prize by showing that the energy in a beam of light is actually contained in lots of small packets called photons, rather than being spread out evenly in the beam. The energy stored in each photon depends on the color (frequency) of the light. Red or infrared light has its energy stored in lots of low energy photons, while blue or ultraviolet light has its energy stored in a smaller number of high energy photons.
Putting it all together...
We know now that electrons absorb or emit fixed sized bursts of energy when they move up or down inside a quantum well. We also know that each color of light is made of packets of energy of a fixed size. If we shine light on a quantum well, then interesting things happen... If the photons in the beam of light contain just the right amount of energy, they can "push" electrons up to higher "rungs" on their ladder of energies! In other words, the quantum well turns light of particular colors into electricity of particular voltages. Scientists and engineers have made many different kinds of light-detecting devices from quantum wells, including solar panels and some modern digital cameras.
Quantum wells work the other way round too. If we attach a battery to the quantum well, then we give lots of electrical energy to the electrons and they are able to "climb" high up their ladder of energies. Over time, they drop down the ladder and release bursts of energy of fixed sizes. This energy is stored inside photons, meaning that a beam of light comes out of the quantum well. Again, many useful devices have been made, including lasers and LEDs.