Tommy
Just to add my two cents to the mix.
The terms Narrow Gauge, Standard Gauge, & Broad Gauge when referring to railroads all share a common term 'Gauge', which refers to the distance between the rails that carry the rolling stock (i.e. locomotives, and the respective railroad cars of various types).
Now, the manner in which this distance (i.e. gauge) between the rails is measured varied from one location to another. As far as I know the two most common methods were; 1) The less common of the two methods, is measuring from the center of the railhead on one side to the center of the railhead on the other side, and 2) The most common method, is measuring from the inside edge of the railhead on one side to the inside edge of the railhead on the other side.
OK, now that we've got the term 'gauge' defined. Where do the terms Narrow, Standard, & Broad fit in? Well, the use of these terms evolved over a period of time.
First off, the vast network of national and international railroads that we're familiar with today didn't exist at the start obviously. In the beginning there were a whole bunch of small individual railroads built by various individuals that had varying ideas of just what the best design was. In the area of gauge the "best choice" was all over the place. Ranging from a gauge "seven feet" to "fifteen inches."
Most of these railroads were built to connect two specific points to each other mainly to move goods from their point of origin to the market place where they could be sold. In many cases the distance between these two points weren't more than a couple of hundred miles, and most were very much less.
As the network of these independent railroads began to develop. One of the problems encountered was the fact that where two railroads of different gauges came together. To move the goods any further they had to be unloaded from one railroad's cars and loaded on the railroad cars of the other. Regardless of the method used, this process took time and labor, which directly translates to an increase in the price of the goods. Another result of this problem was the term "break in gauge" was coined.
The next step in the evolution was that within specific geographical regions a general consensus developed between the various railroads that it would greatly benefit all to settle on a common track gauge. This helped to reduce the vast number of gauges used. However, there were financial, control, and even national security concerns that limited further development of a common gauge being accepted by all the railroads.
Eventually these concerns were addressed in one manner or another and the gauge of 4 feet, 8 and 1/2 inches was settled on as the "best choice" and was called the "Standard Gauge". After the general acceptance of the Standard Gauge, any gauge measuring less was referred to as a Narrow Gauge, and any gauge measuring greater was referred to as a "Broad Gauge"