The track gauge of a railway, the back-to-back distance between the inner edges of the rails, is one of the more critical dimensions of a rail system; systems with gauges that differ are fundamentally incompatible, incapable of interoperability. Certainly, track gauge is not the only parameter essential to such interoperability; loading gauge, the cross-sectional profile within which a rail vehicle must fit, is almost as critical, as are such matters as couplings, signalling systems, and much else. However, none are such a fundamental incompatibility as gauge.

Many different gauges are in use around the world. About 60% of the world's route miles are of the so-called Standard Gauge of 4' 8½"; the rest are either broader or narrower. There are many reasons for this variance worldwide. Moving a rail vehicle between gauges generally required the swapping of wheelsets or trucks, which became a well-practiced art in some cases of trains that had to reguarly undergo a gauge change. More recently, movable wheel systems have been developed, e.g. by the Spanish Talgo company.

A certain taxonomy of track gauges can be seen. While no formal system exists to definitively categorize them, the following is my own system which more-or-less fits in with those I've seen held by others. It also pretty much follows the practice of the builders of locomotives and railway equipment, who generally carried standard designs to fit any gauge within each category. In other words, a manufacturer, say Baldwin, would have a small narrow gauge locomotive suitable for 1'6"-2'6" lines, a larger narrow gauge design suitable for 2'6" - 3'6" lines, and so forth.

Under 1'6" (500mm): Miniature and Model
Few if any railway systems intended for work have been constructed to gauges this small. However, railways for tourists, entertainment rides, and other such applications are regularly built to gauges in this range. Model railways, obviously, also fall into this category. The division betwen miniature and model is not hard and fast, but one possible division is that miniature railways carry human passengers, while model ones cannot.
1'6"-2'6" (500-750mm): Small Narrow Gauge
Much cheaper to build and operate than a full-size railway system, but narrow-gauge railways in this category are just too small to replace a full-size system. They tend to be local in scope and limited in purpose. Many of these systems were built for a specific industry; mining, quarrying, farms, works railway systems, and the like. These include most of the Welsh narrow gauge lines (Ffestiniog, Talyllyn, etc.), which mostly served the slate mining industry. Some were built for general purpose service, but they tended not to last.
2'6"-3'6" (750mm-1000mm): Large Narrow Gauge
Large enough to be a serious proposition as a general-purpose railway system, large narrow gauge lines like these proliferated in areas where the cost of a standard-gauge system would have proved prohibitive. Often, the tighter turns possible in a narrow gauge system reduced the expensive civil engineering works required to an affordable level. These narrow gauge systems were especially popular in mountainous areas, good examples of which are the numerous Swiss narrow gauge systems, most of which still thrive today, and the Colorado narrow gauge lines in the USA. They were also popular in the mining and logging industries, where the tracklaying was required to be cheap and quick, but substantial load-carrying ability was required.
3'6"-4'8½" (1000-1435mm): Narrow Full Size
Gauges in this range were often adopted as the 'full-size gauge' in many regions of the world, particularly Africa and Australia. While narrower than the Standard Gauge, locomotives and cars could attain nearly the same dimensions as true Standard Gauge lines. Gauges in this range were acceptable choices as a full-size gauge.
4'8½" (1435mm): Standard Gauge
The standard track gauge of Europe and North America, and in addition many other areas around the world. While it became 'standard' only because of historical precedent, it certainly seems to be right on the 'sweet spot' between too wide and too narrow. Wider, and minimum radii become too great, and the cost of civil engineering works become intolerable; narrower, and the ability to carry heavy or large loads at speed becomes dangerously compromised through instability.
4'8½"-5'6" (1435-1700mm): Broad Full Size
Standard gauges wider than 4'8½" were adopted in many places, including Spain, Eastern Europe and much of Asia, and parts of Africa and Australia. Permitted trains somewhat larger than Standard Gauge, but in general trains on these railway systems are little larger than the average. Gauges in this range provide an acceptable choice.
Greater than 5'6" (1700mm): Broad Gauge
These gauges are practically extinct. While gauges narrower than the ideal range of somewhere between 3'6" and 5'6" have the temptation of cheaper construction, the same does not hold true for broad gauge. The biggest proponent of a broad gauge was the early Great Western Railway in England, and in particular its chief engineer, Isambard Kingdom Brunel. The GWR at first built to the imposing gauge of 7'0¼" (approx 2140mm), but incompatibility with the Standard Gauge everyone else built to doomed the idea. Other proponents of a broad gauge included some early American railroads (though few built wider than 6') and Hitler's Nazi Germany (which produced grandiose plans for super-wide trains, practically ocean liners of the rails).