Gauge, as in string gauge, is the thickness of the strings. String gauge is usally is measured by the thickness of the thin e string. String gauges tend to be between 0.09 and 0.14 but other gauges and variations can be bought. The most popular gauge of strings for electric guitar is 0.09, acoustic guitars usally use 0.10.
More experienced guitarists may experiment with higher gauges (.011 or .012 and thicker) because they offer better sustain and tone.
Higher gauged strings are harder to play and feel 'heavy' under the fingers. High gauge strings are ideal for down-tuning and many heavy metal bands use thick gauge strings such as 0.11-0.14 mm, but this may require an ajustment of the truss Rod and saddles (to be done by a guitar technician).
The most common string gauges