Hi Easy Rider. Welcome to EpicSki!
"Fall Line" is a common term that goes back a long way in skiing. I don't know who first uttered it, but it stuck, for better or worse. Fear not--it doesn't have anything to do with falling!
"Fall Line" very simply means "straight downhill." Is the direction a ball would start to roll if you placed it on the slope.
It is a significant direction, for many reasons. If you're standing on the slope, and you don't want to move, you must place your skis directly across--at right angles to--the fall line. When you release your edges, your skis will tend to "seek the fall line" as gravity pulls you down the hill.
Good skiers quickly develop a sense of the fall line--the direction gravity tries to pull them at any moment. This sense is critical for choosing line and tactics, controlling speed, and generally feeling at ease on the slopes. Once you understand and can sense the fall line, gravity starts to become your friend!
There is a common misconception about the fall line that you may have heard. The fall line is not "the path a ball will follow as it rolls down the mountain." Think about it--once a ball is moving, it has momentum, and sometimes that momentum will even carry it uphill for a distance--as it rolls up and over bumps and rolls, for example. The fall line is straight downhill from any given point--which is not necessarily the direction the ball will roll as it cruises over that point.
"Fall line skiing" refers to skiing a line more or less directly down the hill, generally involving short, quick turns (or more likely, short, quick braking movements). It is a common expression in mogul skiing, the line that competitive mogul skiers strive to maintain. On the other hand, it is the fastest possible line down any slope, so it is not always the best line. Indeed, I would argue that it is often the worst line you could take--unless you really want to go as fast as you possibly can. Race courses are rarely set "in the fall line," except on very flat sections. And expert skiers--with the possible exception of moguls--tend to make turns that carry them intentionally away from the fall line, in order to control speed without the need to resort to braking and skidding.
Does that answer your questions?