It’s clear how to subdivide a face, but I want to subdivide just an edge. Or, what I really want to do is add a vertex to make a tri a quad, or a quad a pentagon, etc, but still have just one face. Can it be done? Thanks.
Yes, and exactly how you subdivide a face. 
Just select the edge and press the subdivide button or use the specials menu, which ever you prefer.
/Becs
When I select the edge and subdivide it, more faces are also created. If I delete those extra internal edges and try to create just one face (pressing F), it tells me something about an fgon, and doesn’t create anything. What now?
It will create new faces. It replaces the quad with two tris. It can’t make fgons because Blender doesn’t support them yet. Hope this helps
/Becs
To turn a tri into a quad face, subdivide one edge, then select the two resulting tris, and press Shift-J, which will hopefully join the two tris into one quad.
Thanks Matt, that helped a lot. I started with a quad, so I sub-divided an edge, then used Alt-J to join 2 tris into a quad, and was left still with 2 faces, but that’s okay.
I didn’t realize that blender will only let me create 3 or 4 sided faces. The game-engine I’m working on renders many-sided polygons a little faster than tris, so that’s the only reason I’m not going with all tris.
This game will eventually go under OpenGL, and all models can be done with tris.