Robert,

I had a couple of reactions - and therefore will respond a several levels:
What Dimension (plane? 3-Space, abstract?)
What objects (polygons, Polyhedra, other ... )

The usual starting point is plane, convex polygons (e.g. a square, a hexagon ...)
We call the segment (finite - not the infinite line) joining two vertices which do not share an edge a diagonal.
(E.g. a square has two diagonals.)

In 3-space, with a polyhedron (e.g. a cube) we have vertices, edges and faces.
A segment joining two vertices which are not already joined by an edge (adjacent) is also called a diagonal.
E.g. a face diagonal - a diagonal of on of the square faces, or a body diagonal - running through the middle, not across a face.

More generally, mathematicians use that word even in more abstract or unfamiliar situations.
So given a 'graph' - a collecting of vertices and edges, then a segment joining two vertices which is not an edge is also called a diagonal.

Hope this helps!

Walter Whiteley