A tetrahedron consists of 4 triangle faces that join to form a solid figure with 4 faces and 4 vertices. In the case of a regular tetrahedron all the triangles are equilateral.
As far as labeling the faces goes there is no standard way to do that. Each of the faces is accurately identified by the points that lie at the vertices and the format used to represent the faces such that no doubt exists.
Even for a cube there is no mathematical standard notation being used in calling the faces top, bottom, etc., it is just a convenient way of doing so. For a tetrahedron one of the faces on which it sits can be called the base, the other three sides could be referred to as left, right and back.
I don't know about ant standard but if tetrahedron is based on a triangle ABC and has the vertex D then the four faces can be labeled on the basis of the names of the triangles of which they are made of.
The labels would be: ABC, ABD, BCD and CAD.