Hmmm...sounds simple. But it's not quite that simple!
Turn radius does refer to the size of the turn, assuming that it is part of a circle. But few turns are actually circular, so the actual radius typically varies throughout the turn. Still, "short-radius turns" refers to "small" turns, "short" turns, or "tight" turns, as opposed to the larger, slower, turns known as "medium radius" or "long radius."
To me, the important difference is more one of timing than actual size. "Short radius turns" are typically quick and rhythmical, maybe an average of two seconds per turn. Few real "short radius" turns made by expert skiers are as tight in actual size as the turns most beginner skiers make, but at the slow speeds of beginners, it would seem odd to describe them as "short radius turns"--even though it might be accurate.
Anyway, the relation between turn radius and "sidecut radius" is also not as simple as some have suggested either. Sidecut radius refers to the size of the circle that would be made if you continued the curved edge of a ski all the way around. But the sidecut radius of the ski has actually very little to do with the size of a "pure-carved turn" that the ski can make--which can, in fact, vary greatly. In fact, sidecut radius represents the maximum radius a ski could theoretically "pure carve" (no skidding, no slipping, no breaking away of the snow). Most "pure-carved" turns will be smaller than the ski's sidecut radius--often much smaller. Read the thread theRusty linked to in post #4 above, and especially note Tom "PhysicsMan's" formula for calculating "pure-carving" radius, on hard snow, as a function of the sidecut and edge angle to the snow (pure-carve radius = sidecut radius x cosine of the edge angle). This means that a ski with a 12 meter radius sidecut, tipped 30 degrees to the snow, would theoretically "pure-carve" an 8 meter radius turn. At 60 degrees, it would carve 4 meter radius turn! All this is highly theoretical, of course, because there are so many other variables that apply--snow texture, ski torsional (twisting) and lateral (sideways bending) stiffness, amount and distribution of pressure, and skier technique, to name a few.
The "Fall Line" is, simply, "straight down hill" from any given point. It is the direction your parallel skis must be perpendicular to if you want to stand still on the slope. It is the direction a ball would START rolling, if you placed it at any given point on the slope. But it is NOT the path the ball would take all the way to the bottom. Think about it--if you roll a ball down a real slope, with bumps and rolls and such, it will take a curvy path, influenced by the "Fall Line." But it's actual path would be a function of the Fall Line at any given point, and the ball's momentum, and any obstacles that might also deflect its path. Sometimes it would even roll UP hill--up a mogul, for example, or up the side of a gully, all the while "seeking the Fall Line," but never actually going the direction of the Fall Line, except it's initial movement from a stop. (The only exception would be a perfectly smooth and flat slope, like a table tipped up at an angle, where the Fall Line is constant and at the same direction at every point on the slope.)
When we speak of the "Fall Line" part of a turn, we are generally referring to the moment you're traveling straight downhill. We've recently seen some people using the term differently here at EpicSki, but I think there is generally widespread agreement on this definition.
Sorry--things are rarely as simple as we'd sometimes like!