Edge Selection - Selection highlights vertices only? The first picture is showing how the edge selection should look. The second picture is mine, it is showing how my selection looks.

I'm trying to join the bridge of the nose and my selection looks different on
screen, only the vertices seems to be selected but the edges are not.
Looking at your image, you are in vertex selection mode, that doesn't explain why the edge in between the two selected vertices isn't highlighted, so most probably you have doubles. Select all by pressing A once or twice, then  W  and choose remove doubles, watch the status bar at the header of Blender to see how many vertices are removed, this should work normally, you can set selection by edges instead of vertices by pressing Ctrl +  Tab  then choose Edge.