In much of the world, "football" refers to what Americans call soccer:
http://soccernet.espn.go.com/index?cc=5901
(Are you familiar with David Beckham? He plays football according to Europeans, South Americans and Asians. But Americans say that he plays soccer.)
American football is the game you see played on TV during the Super Bowl, with the players wearing helmets and padding, kicking and passing an oblong "pigskin":
http://en.wikipedia.org/wiki/American_football
Rugby is similar to American football in some respects, but there are just as many differences between the two:
http://wildcat.wsc.edu/clubs/rugby/what_is_rugby/