A stellated polyhedron is a polyhedron whose sides are all identical regular polygons or stellated polygons, whose dihedral and solid angles are all identical, and which is self-intersecting. That is, the faces intersect one another.
There are two possible types of stellated polyhedra.
One of these looks much more like most people's impression of a star, and is the result of taking a regular polyhedron (like a dodecahedron) and extending each of its faces from a regular polygon into a stellated polygon. The result is a very spiky, star-like thing. The other is the result of extending the sides but leaving them as pentagons; at a certain size there is a different set of intersections than in the original figure. This result looks like the Alexander's Star puzzle that was one of the zillions of imitators to the Rubik's Cube. It can also be described as an icosahedron with a short pyramidal indentation on each face, but the stellated dodecahedron framework was crucial to Alexander's Star as the exposed parts of the dodecahedral faces were the parts you had to try to match up colors on.