*conductance*is defined as \(G \equiv 1/R\), so \(I = G V\).

In a liquid flow analogy, voltage is like the net pressure across some pipe, current is like the flow rate of liquid through the pipe, and the conductance characterizes how the pipe limits the flow of liquid. For a given pressure difference between the ends of the pipe, there are two ways to lower the flow rate of the liquid: make the pipe longer, and make the pipe narrower. The same idea applies to electrical conductance of some given material - making the material longer or narrower lowers \(G\) (increases \(R\)).

Does anything special happen when the conductance becomes small? What does "small" mean here - small compared to what? (Physicists love dimensionless ratios, where you compare some quantity of interest with some characteristic scale - see here and here. I thought I'd written a long post about this before, but according to google I haven't; something to do in the future.) It turns out that there is a combination of fundamental constants that has the same units as conductance: \(e^2/h\), where \(e\) is the electronic charge and \(h\) is Planck's constant. Interestingly, evaluating this numerically gives a characteristic conductance of about 1/(26 k\(\Omega\)). The fact that \(h\) is in there tells you that this conductance scale is important if quantum effects are relevant to your system (not when you're in the classical limit of, say, a macroscopic, long spool of wire that happens to have \(R \sim 26~\mathrm{k}\Omega\).

Example of a quantum point contact, from here. |

Conductance quantization can happen when you make the conductance approach this characteristic magnitude by having the conductor be very narrow, comparable to the spatial spread of the quantum mechanical electrons. We know electrons are really quantum objects, described by wavefunctions, and those wavefunctions can have some characteristic spatial scale depending on the electronic energy and how tightly the electron is confined. You can then think of the connection between the two conductors like a waveguide, so that only a handful of electronic "modes" or "channels" (compatible with the confinement of the electrons and what the wavefunctions are required to do) actually link the two conductors. (See figure.) Each spatial electronic mode that connects between the two sides has a conductance of \(G_{0} \equiv 2e^{2}/h\), where the 2 comes from the two possible spin states of the electron.

Conductance quantization in a 2d electron system, from here. |

A junction like this in a semiconductor system is called a quantum point contact. In semiconductor devices you can use gate electrodes to confine the electrons, and when the conductance reaches the appropriate spatial scale you can see

*steps*in the conductance near integer multiples of \(G_{0}\), the conductance quantum. A famous example of this is shown in the figure here.
In metals, because the density of (mobile) electrons is very high, the effective wavelength of the electrons is much shorter, comparable to the size of an atom, a fraction of a nanometer. This means that constrictions between pieces of metal have to reach the atomic scale to see anything like conductance quantization. This is, indeed, observed.

For a very readable review of all of this, see this Physics Today article by two of the experimental progenitors of this. Quantized conductance shows up in other situations when only a countable number of electronic states are actually doing the job of carrying current (like along the edges of systems in the quantum Hall regime, or along the edges of 2d topological materials, or in carbon nanotubes).

Note 1: It's really the "confinement so that only a few allowed waves can pass" that gives the quantization here. That means that other confined wave systems can show the analog of this quantization. This is explained in the PT article above, and an example is conduction of heat due to phonons.

Note 2: What about when \(G\) becomes comparable to \(G_{0}\) in a long, but quantum mechanically coherent system? That's a story for another time, and gets into the whole scaling theory of localization.