Ever wondered why it sounds like "sock" but is spelled "soccer"? It's because the British school kids who came up with the name were shortening the word "association". The kind of football that came out of the Rugby School was Rugby Football, and the kind that was managed by The Football Association was Association Football.

Rugby Football became "rugger", Association Football became "assoccer" and eventually "soccer".

Both kinds of football migrated across the Atlantic. The first game played in Toronto was played in 1859, 4 years before the formation of the British Football Association, and the first official Association Football rules. In the USA, "soccer" wasn't officially used for association football until the 1920s. Before that, every regulating association used "football" or "foot ball". Gridiron football obviously comes from Rugby's rules. And, until 1955 the governing body overseeing what eventually became Canada's gridiron football was called the Canadian Rugby Union, long after the rules completely diverged from the "Rugby Football" rules.

Eventually, both sides of the Atlantic dropped the qualifiers for what is now called simply "football". "Rugby football" became "Rugby" basically everywhere. "Association Football" became "Football" in the UK and places where it was the most popular football game, and North America adopted the British slang term "Soccer". The modified version of Rugby Football that used a grid of lines on the field became "Football" in North America, and didn't really get another name elsewhere because it wasn't played anywhere else. Meanwhile, Australia did their typical thing and called their Australian Rules Football game "Footie" and use the British term "soccer" for the Association rules.

Other interesting tidbits:

• Why does American Football sometimes get called "Gridiron" football? Why grid? Why iron? A gridiron is a surface for cooking over a fire, more often called a "grill" today. The term "grid" comes from a gridiron, which is also related to a griddle. Iron doesn't actually mean "iron", it just comes from association with the metal in the word "griderne" which led to gridiron. So, although parallel lines aren't a "grid" in modern language, they look like a grill, which used to be called a "gridiron"
• "Soccer" is still occasionally used in England, like in the TV show Soccer Saturday
• The name "Football" may not come from hitting the ball with a foot. One possible source of the name comes from the game being played on foot. There were early "foot ball" games where it was illegal to kick the ball. This would make sense because the first people to write down the rules for these games were teachers and students at English "Public Schools" (which are the most elite forms of what most of the world would call a private school). They were distinguishing it from ball games played on horseback. But this isn't necessarily clear because the terms "football" "handball" and "hockey" were used as early as 1363.