Gauge is a measure of the diameter of wire or jewelry. For example, gauge is used to describe the thickness of piercings. Measures in gauge can be confusing because, counterintuitively, a higher unit of gauge indicates a smaller diameter. So, 14 gauge piercing is thicker than 20 gauge.
An extremely important point that is often forgotten about gauge is that there are at least two different measures of gauge: the British (Standard Wire Gauge, SWG) and American (American Wire Gauge, AWG) measures. Although these measures are both referred to as "gauge", they indicate different actual widths: for example, 14 gauge SWG is .080 inches, while 14 gauge AWG is .0640 inches.
(I've been informed that there is a third measure of gauge, abreviated BWG, but I don't know what it is used for.)
One might think that it is appropriate to use SWG when measuring wire in Britain, and to use AWG when in America. However, this is wrong. At least in the United States, the appropriate gauge system to use depends on the type of metal you are measuring. Convention dictates to use AWG measurements for non-ferrous wire (copper, aluminum), and to use SWG for ferrous wire (steel).
So, getting 14 gauge steel wire and 14 gauge copper wire from the same company will mean you will have two different thicknesses of wire.
Also note that under AWG, there is a mathematical relationship between the gauge number and the thickness; that is, given one, you can derive the other. As far as I know, there is no system dictating SWG, and the correlation between gauge and thickness seems derived from historical practice.