There are five Platonic solids, all known to the ancient Greeks:
name  face polygon  faces  edges  vertices  faces meeting at each vertex  symmetry group 

tetrahedron  triangle  4  6  4  3  Td 
cube (hexahedron)  square  6  12  8  3  Oh 
octahedron  triangle  8  12  6  4  Oh 
dodecahedron  pentagon  12  30  20  3  Ih 
icosahedron  triangle  20  30  12  5  Ih 
That there are only five such threedimensional solids is easily demonstrated. To have vertices, there must be three of the faces meeting at a point, and the total of their angles must be less than 360 degrees; i.e the corners of the face must be less than 120 degrees: this rules out all the regular polygons except triangles, squares, and pentagons.
Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedra.
Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, (based on ideas regarding the music of the spheres etc.) and identified the five platonic solids with the five planets  Mercury, Venus, Mars, Jupiter, Saturn and the five classical elements. (The Earth, moon and sun were not considered to be planets.)
The shapes are often used to make dice. 6sided dice are very common, but the other numbers are commonly used in roleplaying games.
The tetrahedron, cube, and octahedron, are found naturally in crystal structures.
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