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...
American Wire Gauge
The diameter of corresponding to an AWG size is calculated by this expression:
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%)
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:
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.
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 mm˛ can be found here