a quantum well
Picture a very thin slice of filling in a sandwich — a wafer of one material just a few atoms thick, pressed flat between thick slabs of another. If an electron sits in that thin filling and the surrounding slabs act as walls it cannot easily cross, then the electron is squeezed in one direction (across the thin layer) but stays completely free in the other two (along the plane of the sheet). That is a quantum well: confinement in exactly one dimension. Because the electron is still free to move in a two-dimensional plane, a quantum well is called a two-dimensional structure.
The classic quantum well is a layer of gallium arsenide, only a few nanometres thick, grown between layers of aluminium gallium arsenide, which has a wider energy gap and so forms the walls. The layers are laid down atom by atom by molecular-beam epitaxy, so the interfaces are atomically sharp. The particle-in-a-box rule applies only across the thin direction: the confinement energy rises as 1 over the layer thickness squared, so making the well thinner pushes its levels up and shifts its optical behaviour, while nothing constrains motion within the plane. Stack many such wells and barriers and you build the active region of a real device.
Quantum wells are the workhorse of optoelectronics — they sit at the heart of the semiconductor laser diodes in disc players, fibre-optic transmitters, and laser pointers, and of high-speed transistors. For this structural field the takeaway is the dimensional bookkeeping: a quantum dot confines in all three directions (0D), a quantum wire in two (1D), and a quantum well in one (2D). The number simply counts how many directions have been shrunk to the nanoscale, and hence how many directions of electron motion survive.
A red laser pointer contains a quantum-well laser: a gallium-arsenide-based well perhaps 8 nm thick, sandwiched between wider-gap barrier layers. Electrons and holes trapped in the thin well recombine there and emit light of a wavelength set largely by the well's thickness. Grow the well thinner and the emission shifts toward the blue.
A 2D structure: confined across one thin layer, free in the plane — the basis of the laser diode.
A quantum well is called 2D because two directions of motion remain free, even though the confined layer is a 3D solid. Do not confuse it with a two-dimensional material like graphene: a quantum well is an engineered layer inside a bulk crystal, held by ordinary chemical bonds; graphene is a single atomic sheet held to its neighbours only by weak van der Waals forces.