This information has, of course, been presented many times before, but I had a request from a Forum member for a distilled version. Here's what I told him:
Sure! Glad to help. First thing: Scale and gauge are two different things entirely. All O gauge equipment runs on track 1.25" in gauge, but not all of the equipment is the same size...
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O scale (1:48): Pretty simple and straightforward. 1/4" = 1 foot.
Q scale (1:43): Ever wonder why Corgi, Dinky and other European diecast cars are 1:43 scale instead of 1:48? It's because, in 1:48 scale, 1.25" = 5 foot gauge. If you bump that up to 1:43 scale, the track drops down to 4' 8 1/2" or prototype gauge. 17/64" = 1 foot.
Traditional (O-27): Here, things start getting a little hinky. Traditional scale cars are shrunk to work on the tighter radii of O-27 track. In fact, a lot of S gauge (1:64) modelers will put American Flyer trucks under Lionel traditional size boxcars, et cetera, as they're about the right size and are much cheaper than AF rolling stock.
Semi-scale: Sorta halfway between pure O scale and traditional. Lionel 6464 boxcars are an example of this, and clock in at approximately 1:56 scale.
You'll find similar phenomena with HO/OO and the variety of scales in G gauge.
Hope this helps!
Mitch
EDIT:
Mitch,
Good discussion except that Q scale is actually 1:45 and not 1:43 (4 feet 8.5 inches divided by 45 is 1.25 inches). Lionel's original U.P M10000 passenger train in 1934 was in 1:45 scale and I think the Hiawatha might be the same. "European O-scale" as used in places like France and Germany is 1:45.
The 1:43 stuff comes from "British O-scale" which is 7 mm to the foot or 1:43.5 scale. Our 1.25" gauge track is thus used by 3 different scales.
I sit corrected! 
