hyperplane
A hyperplane through the origin is the kernel of a single nonzero linear functional: the set of vectors v with f(v) = 0. It is as big as a subspace can get without being everything — codimension one, meaning dim V minus one. In R^3 a hyperplane is an ordinary plane; in R^2 it is a line; in R^n it is an (n-1)-dimensional flat slab.
Functionals and hyperplanes are two views of the same data, almost. A nonzero functional f determines its kernel hyperplane, and two functionals have the same kernel exactly when one is a nonzero scalar multiple of the other. So the hyperplanes through the origin are in bijection with directions in the dual space — lines in V* — not with individual functionals.
Sliding the level off zero gives affine hyperplanes: f(v) = c for a constant c is a parallel translate, no longer through the origin. The whole family {f(v) = c : c in F} foliates V into parallel sheets, and f measures which sheet you are on. This is the picture behind 'a functional is a stack of level surfaces' and behind linear constraints and decision boundaries.
Hyperplanes are the geometric face of duality. Separating two sets by a hyperplane is separating them by a functional; a supporting hyperplane of a convex body is a functional maximized on the body's boundary. Every time you see a linear constraint a^T x = b, a margin in a classifier, or a budget line in economics, you are looking at a hyperplane, hence at a covector.
A nonzero functional cuts space into parallel level hyperplanes; its kernel is the one through 0.
Subtle bijection: hyperplanes through 0 correspond to lines in V* (one-dimensional subspaces of the dual), not to single functionals, because f and 2f and -7f all share one kernel. This is the simplest nontrivial case of the annihilator dimension count: a codim-1 subspace has a 1-dimensional annihilator.