Unselectable edge? Help! I don't know if anyone has asked this but I've encountered this problem numerous times on multiple files. When I'm trying to edit a mesh I get this edge that I can't select.

Here's a better picture of what's going on. It's part of the face, but the edge doesn't show up in the wireframe. I tried deleting the face and it put a hole in my mesh.
It looks as though you have a non-planar face.  Since there is no such thing as a curved surface in computer graphics each face gets "triangulated" down to one or more tris (triangles).  So even though a face may only have four edges, it is really composed of two tris, separated by a hidden edge.  This can cause unpredictability when the face is not planar.

To put an editable edge there you can select the two vertices and hit J to join the two vertices.

Alternatively, you can make the face planar by going to Mesh menu > Clean up > Make Planar Faces.