Football to Football: Part 1, Defining the Game and Ball
When talking about “football” and “football” there is a pressing need to distinguish between the two. The definition of “football” in the United States differs from the definition of “football” in the rest of the world. With that in mind, I will make a clear distinction so we avoid confusion as we move forward.
1. football (American):

A. The game itself will be called “football.”
B. The ball itself - will be referred to as “the pigskin,” which is an actual slang term.
1. football (world):

A. The game itself will be called “futbol,” how it is spelled by the Spanish and Portuguese.
B. The ball will be called the “onion.”
To recap:
Game - football and futbol.
Ball - the pigskin and onion.
In context:
“I watched a football game on the TV and Dan passed the pigskin.”
In this context, Dan passed an American football. Passing an American pigskin involves throwing the ball. The pigskin is designed to travel best when it is thrown in a spiral motion, which is difficult to describe, so I let this young man from the University of Kentucky help me. (It’s a video.)
“I watched a futbol game on the TV and Dan passed the onion.”
In this context Dan passed a round football. Passing the onion involves kicking the ball, accomplished through various techniques. Futbol coach Adrian Heath (who is English) demonstrates here.
I realize this might seem rudimentary, but I wanted everyone to be on the same page before things get really in depth. Consider this an introduction, I will use everything at my disposal to explain these two sports, which are near and dear to my heart. Stay tuned!