Platonic solids are convex regular polyhedra (one type of regular polygon and the same number of polygons at each vertex). There are 5 of them.
The Archimedean solids are the polyhedra other than Platonic solids which are convex, have regular faces,
and whose symmetry group is transitive on vertices. There are 13 of them.
Kepler-Poinsot solids are the non-convex regular polyhedra. There are 4 of them.
A prism consists of two copies of a polygon, one a translation of the other, joined by parallelograms on all sides.
In an antiprism, one of the polygons is rotated (in its plane), and triangles join the top and bottom.
Since the base can be an arbitrary polygon, there are infinitely many prisms and antiprisms.
Here we show a few of them with a regular polygon for the base, joined by squares or equilateral triangles.
A Johnson solid is a convex polyhedron where all faces are regular, which is not a Platonic solid, Archimedean solid, prism, or antiprism.
In 1966, Johnson conjectured that there were exactly 92 of them, and gave them the names and numbers below. The conjecture was proved by Zalgaller in 1969. Scroll down for the complete list.
The Catalan solids are the duals of the Archimedean solids. There are 13 of them.