(UPDATE 2): Along with individual vectors, we often need sets of vectors (or points). Many basic sets can be defined according to relative simple equations. The interactive figure below summarizes each of the sets we will discuss.
A unit sphere is defined by the equation $$ \Vert x \Vert_2 = \left(\sum_i |x_i|^2 \right)^{\tfrac{1}{2}} $$ In two dimensions, this is just the Pythagorean theorem. It can be derived for higher dimensions using iterative applications of the Pythagorean theorem. Vectors on the unit sphere are called unit vectors. The set of unit vectors (for a particular norm) is called the unit ball. For a particular norm \(\Vert \cdot \Vert \), this set (in \(\mathbb{R}^n\)) is given by $$ \Ocircle_n = \Big\{ x \in \mathbb{R}^n \ \Big| \Vert x \Vert = 1 \ \Big\} $$
The simplex is defined by the set of vectors with non-negative elements that sum to 1. $$ 1^T x = \math_i x_i = 1, \qquad x_i \geq 0 $$ This set of vectors is the set of finite-dimensional probability distributions and so is often called the probability simplex.
The set of vectors whose absolute values sum to 1 is called the 1-norm ball. $$ \sum_i |x_i| = 1 $$
The unit cube is the set of vectors with elements in the intervals 0 to 1. $$ 0 \leq x_i \leq 1 $$