Pressing Tab button moves curser around the screen

When in a word document or email or even in my browser, when I hit the "Tab" button, the curser then does not advance a space, but will then highlight a link in the document or web page, instead of just advancing with a space. Then if I click on the "space bar" while typing, I cannot advance the cursor. However, in my search for a cause, I found that if I hold down the "fn" key and hit the space bar, it moves.

I know something must be up with shortcuts, and have turned them off, but the problem still comes and goes. I can replicate the problem by repeatedly hitting the tab button.