String gauge is a measure of thickness of guitar strings. This is normally referred to by a numeric value (measured in 1000th of an inch) or by a descriptive phrase (light, extra light, medium etc.). Normally, when using the numeric value, guitarists will refer to the whole set of strings using only the gauge of the lightest string, e.g. 9,10,12 etc.
Electric guitars usually come fitted with gauge 9 strings as standard, though 8-13 gauges are used by many players. Acoustic guitars are generally fitted with gauge 12 strings when built but can be fitted with gauges 10-13.
Different gauge strings can effect tuning, tone and playability. The lighter the string the easier it is to press down or bend; these strings are popular with beginners or players whose style involves a lot of string bends. Heavier strings tend to have a better tone and hold their tuning; these strings are more popular with advanced players, who have more strength in their hands or those who favour tone over playability.
It should be noted that is it usually not possible to change the gauge of strings on a guitar without needing a set-up. This is because different gauge strings are held at a different tensions when in tune which effects the shape of the guitar neck and therefore the action.