Small mistakes happen to the best of modelers. It's good to know how to look at the file and figure out what the problem is so that you can modify the object and fix the problem:
Save a copy of the Suzanne file using the Save As command in the File menu, but increment it, as you learned in the previous chapter, so that you have the original and a copy that is at the current state. This is just to preserve your work, in case you need to go back.
The first thing to check is whether there are two vertices in the same location.
Press G, then Y, and move the eye out in front of Suzanne's face again. While you move it, press the Ctrl button. Make sure that it is far enough from front of her face that the eye does not overlap her skin at all. This time, press the LMB to drop it there.
Press A to deselect all the vertices.
Using the vertex select, click the RMB several times on the vertex that seems to be in both the eye and head. Do you see changes similar to the...