Background

Imagine you were an 19th century engineer and were given the task to sort up among the different sizes wires your employer used. The simplest way would be to use an aritmetic scale: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and so on. This would give you an awful lot of sizes. Worse, 2 is 100% larger than 1, but 9 is only 12,5% larger than 8. Here the mathematicians come to the rescue: Use a geometric series. In a geometric series, the sizes increases with a fixed increment. The simplest geometric series is 1, 2, 4, 8, 16, 32, 64...
 
Although neat and simple, this series is a little bit too coarse for wire size. Two different series were developed, one in the United States in the 1850's and one in France in the 1870's. The former is known as Brown and Sharp gauge, now called American Wire Gauge or AWG for short, the latter is a Reynard series in sq.mm.

American Wire Gauge

The diameter of corresponding to an AWG size is calculated by this expression:
 
D = (92 ^ ((36-AWG)/39))*0.005 inch
 
The higher the number, the smaller the size. Each size is about 25% larger than the previous. This mean moving three sizes doubles the cross sectional area and moving ten sizes, e.g. from 20 to 10 AWG, increases the area about tenfold. Only every other sizes is used in reality. The increase in area between these is about 60%: I.e. 18 AWG is about 60% larger than 20 AWG.
 
Size 0 is often written as 1/0 and the size -1 is written as 2/0, pronounced two-aught. The scale ends with 4/0 AWG

Circular Mils

A circular mil is the area of a circle with the diameter of 1/1000". In practice this number is about a thousand times to small to be usable for wire sizes. Therefore, sizes are usually given in thousands of circular mils, denoted kcmil or previously MCM. One kcmil 0.5067 mm2, which means that for practical purposes the 1 mm2 = 2 kcmil can be used as approximation. (The error is only 1.3%)
 
Kcmil sizes are used instead of AWG for sizes larger than 4/0 AWG. The smallest standard size is 250 kcmil, the largest 2000 kcmil. The sizes follow no obvious logic.

Metric Wire Sizes

The French military engineer Charles Reynard came up with a neater formula than Brown: 10^(n/10) where n=1, 2, 3 and so on. Just like the AWG, each sizes is 25% large than the previous. The neat part is that moving ten steps increases the area excactly tenfold. Normally, only every other size is used. This means you can write the formula as 10^(n/5). The resulting numbers are then:
 
10^(0/5) = 1
10^(1/5) = 1.5848
10^(2/5) = 2.5119
10^(3/5) = 3.9811
10^(4/5) = 6.3096
10^(5/5) = 10
 
In practice these numbers are always rounded. However, for some reason only the sizes from 1.0 mm2 to 25 mm2 follow this logic. Standard sizes up to 1000 mm2 are used, but sizes 35-95 mm2 follow a different series. (See the table below) The metric wire sizes in the electrical industry are always in mm2, never in mm dia. The size of other types of wire, e.g. fence wire, is often given in mm dia.

Japanese sizes

Japan and Korea use a separate system. It appears to have been based on the American Wire Gauge, but the sizes are in sq. mm, rounded and with fewer steps.

Comment

The ampacity of wires depend on a number of factors and converting between metric and AWG sizes is a bit more involved than it seems. Ampacities for wire sizes from 18 AWG - 1000 kcmil and 1.0 - 500 mm² can be found here