Two ways to measure a hole
A ring is a hole, and a hole can be described by how wide it is or by how far it is round. The two are linked by π: a hole 17 mm across is 53.4 mm around. The US and Japanese scales count width; the UK and European scales count distance round. That one fact explains most of what follows, including why the scales never quite agree: steps that are even in width are not even in circumference, and the other way round.
The United States, Canada and Mexico
The American scale is older than any standard and was fixed by the tools jewellers used: size 0 is an inner diameter of 0.458 inches and each whole size adds 0.032 inches. In millimetres that is 11.63 + 0.8128 per size, and rings are made in quarter sizes, each 0.2 mm of diameter apart. One size is 2.55 mm of circumference, the biggest step of the four scales, which is why quarter sizes exist.
The United Kingdom, Ireland, Australia and New Zealand
British sizes are letters, A to Z with halves, and since BS 6820 in 1987 each letter is 1.25 mm of inner circumference with size C fixed at 40 mm. The scale was first defined in 1945 by diameter in sixty-fourths of an inch (BS 1283), and the 1987 switch moved every letter by a fraction of a millimetre, so a ring sized before then may read a half size differently today. Z+1, Z+2 and so on extend the scale for very large sizes.
Europe and the ISO standard
ISO 8653 defines a ring size as the inner circumference in millimetres, and most of Europe simply uses that number: 52, 54, 57. It is the only scale with no conversion in it, which is why our tools pass every size through it. Two older European scales survive on some charts: a Swiss scale equal to the circumference minus 40 (so Swiss 14 is ISO 54) and a German scale by diameter.
Japan and East Asia
Japanese jewellers number rings from 1, at 13.0 mm inner diameter, adding one third of a millimetre per size, so each step is 1.05 mm of circumference, a little less than a UK letter. The same numbers are used in China, Hong Kong, Korea and much of East Asia. Secondary sources disagree about whether the industrial standard JIS S 4700 counts by diameter or by circumference; the charts jewellers print follow the 13 mm plus a third rule, and so does this site, which calls it the trade scale rather than quoting the standard.
India and others
India uses a numbered diameter scale of its own, Italy and Spain count the circumference minus about 40 like the old Swiss scale, and some Latin American countries use the US scale. If a chart gives you millimetres, the chart page’s finder takes a diameter or a circumference directly and skips the naming problem.
Converting safely
Turn the size you have into a circumference using its scale’s rule, then take the nearest size on the scale you want and note the gap. The converter does exactly that and prints the gap in millimetres. A gap under 0.3 mm is nothing; a gap of 0.6 mm or more is a quarter US size, and worth leaning the right way for the band you are buying.