Where did American football get its name?
Meanwhile, American colleges were playing their own version of what they called football -- primarily a kicking game like soccer that allowed more robust hits, like rugby. Every school had its own set of rules, so it was difficult to organize intercollegiate matches. As a result, the first few schools to play "football" sat down and agreed to a standard set of rules, which were modeled after the English rugby code.
At that point, American football could easily have become known as American rugby, but since players and fans had already been calling the game football, the name stuck. The name, in any event, wasn't so peculiar as we might consider it today, since the game was much more kicking-oriented than it is now. Kicks were allowed from any point on the field (now they're limited to being taken from behind the line of scrimmage); field goals were initially worth more than touchdowns; and since there was no forward pass, teams would often resort to sacrificing possession by punting deep and going on defense, especially if their running game was getting bogged down.
So while it might seem odd to modern viewers to refer call a game "football" that doesn't primarily employ kicking, there is a legitimate historical reason for it. Also keep in mind that of the world's six major football codes -- association football, rugby union, rugby league, Gaelic football, Australian rules football, and American football -- all of them except association football permit use of the hands in open play, and all of them involve using the foot to a greater or lesser extent to propel the ball.
Most of the world refers to American football as "gridiron" football, because of all the line markers on the field. That's certainly a better and more inclusive term than "American football," since Canadians play a variety of the same game. "Gridiron" can also be used to encompass other variations on the game, such as arena football.