The fall line is a line straight down the hill, pointing towards the direction that a ball would take if let loose. It is used also to denote the middle of the turn or the apex, as most turns tend to be across the hill and when the skis are pointing down, we refer to the skis as "being in the fall line" - frame 6 below:
The fall line keeps changing with the terrain, it is not always "down towards the bottom", as the slope may have features like rollers and "fall aways".
Also, the apex of the turn is not always in the fall line, although that is a good approximation and many tend to refer to both as being at the same time (we are guilty of that as well, sometimes).
Don't confuse, however, being in the fall line with being "at the gate" in racing. Being in the fall line simply means skis pointing straight down the hill, which on a course usually happens somewhere above the gate.
Sometimes you will hear the imaginary line drawn from the gate down referred to as the fall line, i.e. the opposite of the riseline++ but that's infrequent and not what most mean by it. However, the reverse is true - the "rise line" is the line pointing straight up the hill from the gate.
Turning the skis into the fall line is when they tend to pick up speed, as they glide straight down the hill. When the skis turn across the hill, they tend to loose speed, so a very effective speed control measure is to keep the skis pointing down as little as possible and pointing or turning to the side as much as possible, basically doing short turns or travelling across the hill more (offset turns).