Gauge is the term used in railway parlance to explain the distance between the running rails of a railway track. In the UK, the gauge used on standard railways is 4ft 8 1/2 in or 143cm. The same term is also used when referring to a model railway.
The term 'gauge' in model railway terms refers to the distance between the running rails, not the scale of the trains which run on the rails. The most popular for model railways in the UK is OO gauge, which has a rail separation (gauge) of 16.5mm. Other popular gauges are O gauge and N gauge, with 32mm and 9mm gauges respectively.
Railways employ two gauge standards: standard and narrow. Standard gauge is by far the most popular, used for all mainline and secondary routes in the UK, with a gauge of 4ft 8 1/2in. Any track in the UK with a smaller gauge - often used around industrial sites - is referred to as 'narrow gauge'. Wider track gauges were used in the UK during the pioneering railways such as 5ft 6in on the Dundee and Arbroath Railway (1836-1847) and the Arbroath and Forfar Railway, with 7ft 1/2in gauge being used by Brunel across many routes, too, until converted to 4ft 8 1/2in gauge in 1892. These wider gauges are referred to as 'broad gauge'.
The same terms are applicable to a model railway. A piece of model railway track can be used for different scales. For instance, common OO gauge (16.5mm) track can be used to depict standard gauge railways 4mm:1ft scale, or narrow gauge railways if the models on it are larger (On30 or O-16.5 being the most common to model 2ft, 2ft 6in or 3ft gauge narrow gauge prototypes).
Modellers often change track gauges to better match the scale of a prototype, for instance P4 or EM, both of which are gauge-widened equivalents of 4mm:1ft scale modelling, as used in OO gauge. Modellers commonly mis-use the term 'gauge' to describe 'scale', particularly when describing layout accessories or die-cast vehicles suitable for use on model railways. For this reason, it's important to differentiate between gauge and scale.