a polytope
A polygon is a flat shape with straight edges — a triangle, a pentagon. A polyhedron is its 3D cousin with flat faces — a cube, a pyramid. A polytope is the same idea carried into any number of dimensions: a shape bounded by flat pieces. In 2D a polytope is a polygon, in 3D a polyhedron, in 4D and beyond it is a 'polychoron' or simply an n-dimensional polytope. It is the natural higher-dimensional home for the most basic flat-sided shapes.
There are two equivalent ways to pin a (convex) polytope down, and their equivalence is a small marvel. The first is from the outside in: a polytope is the set of solutions to a finite system of linear inequalities — slice space with finitely many flat cuts (half-spaces) and keep the bounded region they enclose. The second is from the inside out: a polytope is the convex hull of a finite set of points, the smallest dent-free shape wrapping a handful of corners. That these descriptions coincide (the Minkowski-Weyl theorem) means a polytope can be presented either by its bounding inequalities or by its vertices, and switching between the two is a deep practical problem. A polytope's flat pieces of every dimension are its faces: vertices (0-dimensional), edges (1-dimensional), then 2-faces, and so on up to facets (the top-dimensional faces).
Polytopes are the shape of linear programming — every linear optimization problem is a search over the vertices of a polytope, and the simplex method literally walks edge to edge along one. The face-counts obey a beautiful pattern: in 3D, Euler's relation V - E + F = 2 connects vertices, edges, and faces of any convex polyhedron, and it generalizes to an alternating-sum identity (the Euler-Poincare relation) in every dimension. One honest note: 'polytope' usually means the CONVEX kind, but the word is also stretched to non-convex and even abstract combinatorial polytopes; and visual intuition runs out past 3D, so in four dimensions and up we trust the algebra of inequalities and vertices, not a picture in the mind's eye.
A cube is the polytope defined by six inequalities: 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1. Equivalently it is the convex hull of its eight corner points (0,0,0), (1,0,0), ..., (1,1,1). Count its faces: V = 8 vertices, E = 12 edges, F = 6 square facets. Check Euler's relation: 8 - 12 + 6 = 2. The two descriptions (inequalities versus vertices) describe the very same solid.
One polytope, two faces of the same coin: a system of inequalities and a list of vertices.
Visual intuition stops working past three dimensions, but the definitions do not. A 4-dimensional polytope is perfectly rigorous as a system of inequalities or a convex hull, even though no one can truly 'see' it; trust the algebra, not the mind's eye.