The dodecahedron has twelve regular pentagonal faces, twenty vertices and thirty edges, with three faces meeting at each vertex. Euler's formula gives 20 - 30 + 12 = 2. Its dihedral angle is about 116.57 degrees, the angle whose cosine is minus one over the square root of five.
It is the dual of the icosahedron: twelve faces and twenty vertices against twenty faces and twelve vertices, with thirty edges in common. Joining the centres of one solid's faces produces the other, in either direction, and the two share a symmetry group of 120 operations, 60 of which are rotations.
The golden ratio enters through the pentagon. In a regular pentagon the diagonal is longer than the side by exactly that ratio, so any solid with regular pentagonal faces carries it throughout. A less obvious consequence is that a cube can be inscribed in a dodecahedron with its twelve edges lying as diagonals across the twelve pentagonal faces, and there are five distinct ways to do it. Those five inscribed cubes are the reason the rotation group of the dodecahedron is the group of even permutations of five objects: the rotations shuffle the cubes.
Euclid constructs the dodecahedron in Book XIII of the Elements, and it is the hardest of the five to build with compass and straightedge, which is part of why it comes last. The book then compares the edges of all five solids and closes by arguing that no further regular solid can exist.
A caution about crystals. Pyrite grows twelve-sided crystals that are often called dodecahedra, and they are not regular ones. The pyritohedron has twelve pentagonal faces, but the pentagons are irregular and the solid has no fivefold axis of symmetry. It cannot have one: a true fivefold axis is impossible in any periodically repeating crystal lattice, which is a theorem of crystallography rather than an accident of growth. Photographs captioned as natural Platonic dodecahedra are almost always pyrite.