A norm is a function from the vector space to the nonnegative reals, usually marked by some variant of ||x||, with these properties:

It is very easy to create a new norm in R^n: just take as your unit ball B some bounded (under any old norm) convex open neighborhood of 0 which is symmetric (in the sense that B=-B, i.e. x is in B iff -x is in B). Then for any vector x, let t be the supremum of the reals with the property that tx is in B (t is not attained, since B is open; it is positive, since B is a neighborhood of 0, and it is not infinity, since B is bounded). Define ||x|| = 1/t. The triangle inequality is due to convexity; central symmetry is required to be able to compare ||x| with ||-x||.

To get the standard euclidian l2 norm, take as B a ball; to get the l_infinity norm, take as B a cube; to get the "Manhattan" l1 norm, take as B the generalised octahedron { (x1,...,xn) : |x1|+...+|xn| ≤ 1}.

Any normed space is a metric space. To get a metric space from a normed space, define d(x,y) = ||x-y||.