Scales & Gauges
The model railroading community is divided up into different scales (think neighborhoods) and gauges, which describe not only the size of the equipment (compared to the prototype) they run, but also how wide the track would be if it were a full sized railroad.
Wikipedia has a nice graph explaining the different scales & gauges here.
Common scales & gauges include:
"G Gauge" - a collection of scales including 1/20, 1/20.5, 1/22.5, 1/24, 1/29 and 1/32 designed to run on the the track originally created by the LGB company in Germany. The track width itself is the same for all the scales, with the differing width representing either normal gauge (1/32nd or #1 scale) or some form of narrow gauge track width (20in, 2ft, 30in, 3ft...). These are the scales normally seen in Garden Railways. Several of the above scales are being formalized with their own letters, the most common of which is "F" at 1/20.32.
O scale - 1/48th scale (1/4 inch to the foot). Originally the scale of Lionel & American Flyer with it's three rail AC power "tinplate" track, O scale is making a rennaisance as the baby boomers age and discover that larger can be easier to handle. Companies such as Atlas O and MTH are producing high-quality O Scale locomotives and equipment that generally comes in two versions - three rail AC and two rail DC.
S scale - 1/64th scale. Formerly home only to the American Flyer tinplate collectors, S scale, like O scale, has been making a rennaisance recently, and probably for the same same reasons - HO just seems a little small for some people. S Scale is fortunate in that many of the diecast automobiles you can buy at stores like Wal-Mart are 1/64th scale. Accurate and detailed S scale equipment is now being produced by companies such as Des Plaines Hobbies.
HO Scale - 1/87th scale. The most popular (by a lot) model railroad scale, HO scale has been around since just after WWII. The "HO" originally stood for "Half-O" although it isn't quite that. It is generally the easiest scale for which to find anything you are looking for. A recent development in HO scale is that of the Free-mo modular system, which allows large, interesting layouts to be assembled by many indviduals.
N Scale - 1/160th scale. Showing up in the late 1960s/early 1970s, N Scale is approximately half the size of HO scale, which made it popular among people who either had a limited amount of space or wanted to run prototypical length trains without having to build their layouts in an aircraft hanger. Like HO, there are many many manufacturers of quality train and scenery components in this scale. N scale was popularized by the introduction of the N-Trak modular standards, which allowed different people around the world to build small sections of railroads and then hook them all together at a meet. Fantastically large layouts are able to be assembled, temporarily, in this manner.
Z Scale - 1/220th scale. A recent newcomer, manufacuturers such as Micro-Trains, American Z Trains and Marklin are introducing new locomotives and cars at regular intervals. Within the last two years Micro-Trains has produced US prototype "hood" style diesels such as the GP35 and the GP9 - and an SD40-2 model is on it's way as well. People have created entire Z scale layouts inside of briefcases. Like HO and N scale, Z scale also has a modular standard known as Z-Bend. There were multiple Z-Bend layouts at the 2008 National Train Show.
Now we've gotten scale out of the way, let's talk "gauge."
Often, when talking about model trains, people will say that they model in something like "HOn3" or "On30" or such. What the "nXX" defines is the gauge they model within that scale - normally narrow guage. Narrow gauge prototypes were normally (but not exclusively) set up in rugged regions where the ability to take up less space and turn tighter curves were prized. Common narrow gauge widths included 24 inches (such as the Maine "two footers"), and 36 inches (or 3 feet, common out west). Since HO scale track is nearly a perfect 30 inches in O scale, it is now being commonly used in On30 (narrow gauge, 30 inches) modeling - a quickly growing scale and gauge. If you see a single digit, like a 2 or 3, it indicates feet, where two digits, such as 24 or 30, indicate inches.
Fn3 is 1:20.32 scale equipment being run on G gauge track, to emulate 3 foot narrow gauage. It is somewhat unique in that it's a scale that generally does not have a standard gauge - only narrow gauge.