A narrow gauge railroad is one built to a track gauge less than the Standard Gauge of 4'8½" between the rails. In some parts of the world, however, a gauge less than that is the local standard, and thus the definition I'm going to use here is a track gauge significantly less than the local standard, full size gauge.
Why build a narrow gauge railroad? In a word, it's cheaper. It's partly cheaper because everything is generally smaller; smaller locomotives, smaller cars, lighter rail. Most of these advantages, however, can be had with a standard gauge railroad, which does not have to be built to full-size, main-line standards (although admittedly one would lose all the interoperability benefits). The other advantage of narrow gauge railroads is that the minimum radius of curves that can be traversed is much less. This translates into significantly reduced civil engineering costs, since a narrow gauge railroad can hug the terrain rather than having to bash straight through as a standard gauge system must.
It's not surprising, therefore, that narrow gauge systems have been most popular in mountainous areas. On flat ground, most of the advantages vanish.
However, a case could certainly be made that narrow gauge railroads were almost universally a bad idea. Often the initial savings were smaller than was anticipated, and in the long run the extra cost of running a separate, incompatible railway system dwarfed the initial savings. As a temporary measure, a way to build a railroad cheaply for some immediate need such as mineral traffic or a logging railroad, they were fairly successful for a short time. This kind of temporary railroad has almost completely died out since the diesel-engined truck became tough and cheap enough after the second world war, however.
That the narrow gauge railroad is a concept whose time has passed in North America is best shown by the fact that only one narrow gauge railroad survives in commercial service in the United States, that being US Gypsum's operation in Plaster City, CA. All other surviving narrow gauge railroads are tourist rides only. In Canada, a system survives in Newfoundland I believe.
Worldwide, the surviving narrow gauge railroads are also few. Switzerland is the only country that has a truly successful narrow gauge system; the Swiss narrow gauge lines cover a fairly large percentage of the country, serving most of the Alpine regions and their ski resorts. In the main, they survive on passenger traffic, though the larger routes such as the RhB also carry freight. The Swiss lines are mostly metre gauge systems.
Narrow gauge systems also survive to a lesser degree in the Alpine areas of Italy and Austria, and some in the former East Germany and Poland, although the scale of operations is much reduced from that in Communist times.
Elsewhere in the world, sugar cane plantations keep alive most of the rest of the surviving narrow gauge railroads, in Australia, Indonesia and elsewhere.
Many narrow gauge systems, though obsolete commercially, still operate in a tourist capacity. I can't hope to cover them all in this writeup, of course!
In the United States, the premier surviving narrow gauge systems are the East Broad Top in Pennsylvania, a former coal-hauling railroad, and two surviving stretches of the Denver & Rio Grande system in Colorado, the Durango & Silverton Railroad and the Cumbres & Toltec Scenic Railroad. Also in Colorado, there is the Georgetown Loop Railroad. In California, several former narrow gauge logging railroads operate in tourist service, including the Roaring Camp & Big Trees Narrow Gauge Railroad and the Yosemite Mountain - Sugar Pine Railroad. Narrow gauge trains also run at several California theme parks, including Knott's Berry Farm (running authentic Colorado narrow gauge equipment) and Disneyland (albeit much 'dressed up' from the locomotives' more workaday origins). Real narrow gauge steam locomotives imported from Mexico also run at Walt Disney World in Florida, again in gaudy Wild West dress. In Maine, the Maine Narrow Gauge Railroad runs authentic Maine Two-Footer equipment, much smaller than most American narrow gauge.
Also still surviving in Alaska is the White Pass & Yukon Route, which is rather different from most of the preceding since it survived into the 1980s as a functioning commercial enterprise. Construction of new roads and the closing of ore-shipping customers doomed it, but it found a new lease of life shortly afterwards as a tourist railroad. Although it has one functioning restored steam locomotive, much of the line's traffic is hauled by its fairly modern fleet of brightly-painted diesel locomotives.
In the United Kingdom, most of the surviving narrow gauge lines are in mountainous Wales, including the famous Ffestiniog Railway and the Talyllyn Railway, both constructed to serve slate mines high in the mountains.
Tourist-oriented narrow gauge railroads also exist in much of the rest of Europe, including France, Germany, Portugal and Austria. I hope to come back and cover these in more detail later, either here or on another page.