The fall line is an important concept in skiing and boarding. It is the line directly downhill from the point at which you are standing. Think of the hill as a tilted surface. If you put a ball, or empty a bucket of water, at the top of the surface it will take the shortest, most direct line to the bottom. This is the fall line. The fall line will change direction with the dips and rolls of the mountain. Although the absolute direction will change, it is always directly downhill from the point at which you are standing. The fall line is always determined relative to the observer.
A good ski & board run heads pretty much straight down the fall line. This allows rhythmic turning in both directions. Some runs are across the fall line to various degrees. If you turn equally in both directions you will eventually end up at one side of the run after a few turns. To combat this you have to make turns of different lengths (or shapes) in each direction. This takes a bit of experience to do well.
"Across the fall line" is an important concept. If your skis or board are parallel with the fall line you will take off downhill. If they are perpendicular (at 90 degrees) to the fall line they are "across the fall line" and will not slide as easily. If you dig your uphill edges into the snow they should not slide at all. For skiers, if you are clicking back into your skis (or bindings) make sure that they are across the fall line. If they are not, as soon as you click in you will take off down the hill to a new disaster. It also helps to click into your lower ski first if you have both skis off, or if it is your lower ski that has come off, turn 180 degrees so that your ski that was your upper ski is now the lower ski.