Introduction to Polytopes (Rudimentary)
In the next version of the website, I hope to have a more complete introduction to polytopes. I hope this version helps to find and understand the key definitions.
convex hull: Intuitively, the convex hull of a given set of points is the set of all the points “inside” the given boundary points. Formally, if the coordinates of the points are given as 
then the convex hull is the set of all points with coordinates specified by 
where 
For pictures and further description, see http://en.wikipedia.org/wiki/Convex_hull
polytope: A polytope is defined to be the convex hull of a finite set of point in d-space. For 2-space, polytopes are simply polygons. For 3-space, they include cubes, pyramids, octahedra, etc.
For further description, see http://en.wikipedia.org/wiki/Polytope
face, facet: The boundary of each polytope is comprised of polytopes of smaller dimension. I use the term “face” to refer to a boundary polytope of any dimension. For a d-dimensional polytope, the term “facet” always refers to a (d-1)-dimensional face.
face lattice: Roughly speaking, the “face lattice” of a polytope is the collection of all the lower dimensional polytopes that form its boundary, together with the partial ordering induced by set inclusion. The face lattice poset can be considered purely as a combinatorial object. It represents a “blueprint” for the polytope with all the geometric details (e.g. angles, lengths, etc.) omitted. Two polytopes are “combinatorially equivalent” or in the same “combinatorial class” if they have isomorphic face lattices.