The tetrahedron has four triangular faces, four vertices and six edges, with three faces meeting at each vertex. It is the smallest number of flat faces that can enclose a volume: three planes cannot close a space, so no simpler polyhedron exists. The counts satisfy Euler's formula for convex polyhedra, V, E + F = 2, which here gives 4, 6 + 4 = 2.
It is the only one of the five regular solids that is self-dual. Take the centre of each of its four faces, join the neighbouring centres, and the figure that appears is another tetrahedron, smaller and inverted. The other four pair off with each other instead: the cube and the octahedron are duals, and so are the dodecahedron and the icosahedron.
That there are exactly five convex regular solids and no more is not a tradition or a claim. It is a theorem, and the argument is short enough to give in full. At least three faces must meet at every vertex, and the angles gathered there must add to less than 360 degrees or the surface will not fold into a corner. Equilateral triangles have 60 degree angles, so three, four or five of them fit: that gives the tetrahedron, the octahedron and the icosahedron. Squares have 90 degree angles, so only three fit: the cube. Regular pentagons have 108 degree angles, so only three fit: the dodecahedron. Regular hexagons have 120 degree angles, and three of those already lie flat and tile the plane. Nothing with more sides can be used at all. Five arrangements, and the list closes.
Euclid sets this out at the end of the Elements, in Book XIII, which constructs the five figures and then argues that no other is possible. A scholium attached to that book credits the cube, the pyramid and the dodecahedron to the Pythagoreans, and the octahedron and the icosahedron to Theaetetus, who is generally taken to have written the first systematic treatment of all five. Plato, whose name the solids carry, did not discover them.