Nelson, in Europe there are still quite big narrow gauge railroads alive and in business.
Because of the metric system in Continental Europe most French, German and Swiss narrow gauge systems were in meter gauge (1000 millimetres), in Austria however they used quite a lot of 750 millimeter track (which is almost equal to 2 1/2 ft).
I believe the Swiss Rhaetian Railways operate the biggest narrow gauge system in Europe. It is connected to other systems in the Alps like e.g. the Furka-Oberalp Ry.
Was it only used in logging, mining and urban transport?
No, not at all. The mentioned lines are full-fledged railroad systems, both for passenger and freight. Passenger traffic (tourists!) was and is very important.
An example is the famous 'Glacier Express' luxury train which runs daily in 8 hours over 180 miles (280 km) through 91 tunnels and 291 bridges from Zermatt to St. Moritz (both very fashionable tourist resorts). There are also lots of other name trains in the Swiss Alps, like the Golden Pass Express, Panoramic Express etc.
The main reason for building narrow gauge lines was cost, like Vic said. On the other hand, in the narrow valleys the curve radii just had to be so tight that the construction of a standard gauge track simply was impossible.
Ron
The picture shows the Glacier Express on one of the most spectacular viaducts over the Landwasser river.