The old British system of knitting needle sizes was based on the Standard Wire Gauge which is an indirect measuring system, the higher the number the thinner the needle. Available sizes were usually from about size 14 up to about size 4, occasionally thicker or thinner.
In Europe needles are measured by their diameter, in milllimeters (metric system) and that was adopted in UK several decades ago, but USA is only just beginning to use the metric system and most American knitters still think in terms of the old US sizes.
There was no universal standard to which American knitting needles were manufactured and sizes do vary a bit from maufacturer to manufacturer, but the big difference to the British system is that a direct measuring system was (and still is) in use, with high numbers for the thicker needles and low numbers for the finer ones. American knitting needles are not American Wire Gauge, that like Standard Wire Gauge is an indirect system with higher numbers representing thinner wires. I suspect that most American needles are actually now manufactured to metric sizes and each munufacturer then decides on the nearest USA needle size.
Straight needles with a knob on the ends generally have the size, in whatever format, stamped into the knob. Double pointed needles and circular needles when new may have it printed along the length but it soon wears off, so a needle gauge is a useful tool. The smallest hole into which a needle will fit indicates its size. Metric sizes can also be, very accurately, measured using a calliper tool across the diameter of the needle.
Plastic measuring gauge with UK Imperial sizes and metric on one side, and US sizes and metric on the other side. | Metal "Bell gauge" with UK Imperial sizes. It is the size of the slots, not the size of the holes , which is used to measure a needle. |
Metric/UK/USA needle comparison