Time for action – controlling the visibility of vertices
As horrible as this looks, it's not bad. Only selecting the visible vertices gives you one way to control the vertices on which you are working. It's a tool, but it's no good unless you have control over it:
Press Ctrl + Z to undo deleting Suzanne's vertices.
Press 1 on the NumPad to select the Front View.
Now, look at the 3D View header. There is a button that controls the visibility of the hidden and back-facing vertices, edges, and faces. It is highlighted on the right-hand side of the following illustration. It is just to the right of the three buttons that control whether you are working on vertices, edges, or faces. Left-click on it so that it turns light grey. You'll notice that you can see more edges within Suzanne.
Press A to deselect all the vertices.
Use the Border Select tool to select all of the vertices.
Press X. A menu pops up. Select Vertices.
What just happened?
Okay, everything worked as expected. You selected all the...