What joins one vertex to another and one face to another which is usually by a line segment called?

The term your teacher is looking for is "edge". The edge is a line segment joining two adjacent vertices of a 3D polyhedron. The edge also connects two adjacent faces. Through Euler's polyhedron formula, we can say

V+F-E = 2

which allows us to connect the number of vertices (V) with the number of faces (F) and the number of edges (E) all in one compact formula. If we know two of the variables, then we can solve for the third. 

Igotchu on that if you still need it