When two things are parallel in geometry, they have will never intercept (unless they're identical.) This can be expresses as having equivalent slopes.
The term parallel is usually applied to two lines or occcationally two planes, but there's no reason not to use it for many-dimensional vector spaces. To find if two vector spaces (such as a line or a plane) are parallel in n-space, write them as equations and solve the equation. If the system has no solution that means the spaces never intersect are thus parallel.
See also: Perpendicular