I've had this cursor for some time now. What is it and how can I make it go away, and get the normal cursor with which I can select objects?
If the cursor shown in your picture is moving when you try to select something,
then try to select something with the other mouse button.
Select with right, move 3D cursor with left.
Changed in UserPreferences:
Select with left, move 3D with right.
So I phrase, "the other mouse button".
The most easily available description of the cursor in your picture is the description of the tag you used. Read up on it. (And as cegaton said, "make it your friend" instead of fighting it.)
"The 3D Cursor is simply a point in 3D space which can be used for a number of purposes." this is just a quote from here (read it): https://docs.blender.org/manual/en/dev/editors/3dview/3d_cursor.html