The octahedron has eight triangular faces, six vertices and twelve edges, with four faces meeting at each vertex. Euler's formula gives 6 - 12 + 8 = 2. The easiest way to see the shape is as two square pyramids joined base to base, and that hidden square is one of three mutually perpendicular squares that can be cut through the solid.
It is the dual of the cube, and the relationship is worth stating precisely rather than waving at. The cube has six faces and eight vertices; the octahedron has eight faces and six vertices; both have twelve edges. Join the centres of the cube's six square faces and an octahedron appears inside it. Join the centres of the octahedron's eight triangular faces and a cube appears inside that. The counts swap, the edges stay put, and the two solids share a single symmetry group of 48 operations, of which 24 are rotations. This is why a cube and an octahedron are, from the point of view of symmetry, one object seen from two sides.
Its dihedral angle, the angle between two faces along a shared edge, is about 109.47 degrees, more exactly the angle whose cosine is minus one third. That figure reappears in chemistry as the bond angle of a tetrahedral molecule, and the repetition is not coincidence: it is the angle subtended at the centre of a regular tetrahedron by two of its corners, and the same geometry underlies both.
Among the five, the octahedron sits in the middle almost everywhere. It has more faces than the tetrahedron and fewer than the icosahedron. More faces meet at its vertices than at the cube's, fewer than at the icosahedron's. Its dihedral angle falls between theirs. It is also the one of the five that most readily balances on a point, since its opposite vertices are directly aligned. All of that is genuine geometry, and it is why the shape reads as balance to anyone building a symbolic vocabulary out of the solids. The symbolic reading itself is a modern habit rather than a classical one.