What is this jagged mess between the edges and how do I fix it?
Looks like you are in edit mode of one mesh object, while also having another mesh object below. Your vertices are intersecting the second mesh surface, so your graphics card is drawing the intersection the best it can, and it is looking a little jagged. Try going into local view with the / key on the numpad and see if that helps.
I think I just figured it out. There’s only one object and it has a subdivision surface. Once I clicked on the buttons ‘Use modifier while in edit mode’ and the ‘Apply Modifier to editing cage during edit mode’ it solved it.
Nevermind, that wasn’t it. Weird, I’ll have to google it.
Here is a better picture. I’ve created a new scene, added suzanne, added a subdivision surface set to level 2, and smooth shading
^ That is due to the subdivision modifier smoothing out the shape of the monkey head, so that it is smoothed over the vertices.
To fix that sort of problem, select an edge, and press ctrl-E and move the mouse. This creates a crease, which lessens or increases the way in which the subdiv modifier smooths it.
In this case, if you really wanted to get rid of the higher bits, select the edges underneath them, press ctrl E and then move the mouse until the edges are highlighted pink.
The last image you are showing has certainly not activated the ‘Apply Modifier to editing cage during edit mode’ button. This option is exactly what solves your problem, as the egdes are then following exactly the surface of te subdivided mesh.In your case you see original mesh’s edges with the subdivided surface below.
Why did you think that enabling the option didn’t solve your problem ?
It does, but if you try doing something like extruding a face, you get strange results.
What exactly do you mean by ‘strange results’. Editing with subsurface modifier applied is a bit different than normal editing and can easily confuse if you are not used to it.
ye this is how editing with subsurf is ment to work… you have to use loop cuts and fiddling to get what you want 
you just need to get used to modeling with subdivision surfaces. these are not blender based tutorials but the concept are extremly transferable as most programs out there support Catmull-Clark subdivision surfaces except the just go under different names.
Grant Warwick - hard surface fundamentals- it for max but its really basic and you should be able to transfer the concepts to Blender
Richard Culver - fundamental of subpatch modeling-he wrote this for lightwave but again it covers the basics and is transferable to Blender
if you want to do high-poly stuff to bake normal maps from or as final models than you really need to learn how to work with subdivision modifiers.
I believe the [Set Smooth] buton might have something to do with it too.
I figured it out! If you uncheck the button ‘Limit selection to visible’ you still have the “crease” from before, but now you can see the edges and vertices through it.