Line in space
A point plus a direction vector, swept by a parameter: \mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}.
A line in space is described by picking one point P_0 on it, one direction vector \mathbf{v} to travel along, and letting a parameter t sweep through every point that direction reaches: \mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}. The description is never unique — any point on the line can serve as P_0, and any nonzero scalar multiple of \mathbf{v} describes the same line at a different speed — but the direction vector itself, up to that scalar multiple, is fixed: two lines are parallel exactly when their direction vectors are scalar multiples of each other, and two lines are skew when they are neither parallel nor intersecting, which is the norm in 3D rather than the exception.
Examples
The line through P_1=(1,-2,3) and P_2=(2,3,0) has direction \langle 1,5,-3\rangle and equations x=1+t,\ y=-2+5t,\ z=3-3t.
Air traffic control tracks a flight as a line in exactly this form — a starting position and a velocity vector, evaluated at the current time.