Imagine a piece of pure semiconductor. Electrons in this bulk are free to move around, in three dimensions. In solid state physics, these electrons are sometimes referred to as an electron gas.
It is possible to constrain the movement of the electrons by disallowing one dimension -- that is, by reducing one degree of freedom. The electrons can then move freely in only two dimensions. A system of electrons confined to a plane is called a two-dimensional electron gas, or 2DEG.
Quantum wires have only 1 degree of freedom, and quantum dots have 0 degrees of freedom.
It is important to note that confinement is not perfect. A quantum dot is often defined as confinement to a very small volume of space, such as a sphere that has a diameter on the order of the material's Exciton Bohr Radius. At this size the energy bands of the semiconductor are no longer continuous, which results in the confinement.
Quantum confinement gives rise to many interesting effects, and these effects can be exploited to develop new semiconductor devices. One such type of device are mesoscopic devices, such as single-electron transistors.
Answered by:
Aman Ahuja, Physics Ugrad Student, WPI

'A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.'