The sport now known as American football or gridiron football changed drastically in the past century or two. When the game first gained popularity in North America, it started as rugby football and the ball was generally advanced by kicking it. Consequently, the term "football" had caught on during the sport's early days in North America.

It's also worth noting that were a bunch of different sports that had names which were variations of "_____ football" in the same manner as rugby football. One of those was association football, the full name for the sport known as soccer in the US and "regular" football elsewhere. Where did the term "soccer" come from? In the late 19th century, some slang was developed at Oxford University in England that informally used "_er" as a suffix at the end of words. For example, a five pound note would be called a "fiver" in this slang. These Englishmen applied this slang convention to the term association football, shortening it to assoccer. Over time, this got shortened from assoccer to just soccer. I'm not sure how, but this term made it from Europe to North America, where it ended up being adopted.

Because soccer is short for “Association Football” and was what it was commonly known as worldwide to differentiate it from the other footballs like Rugby football. As some point the UK started just referring to it as “football” and the yanks didn’t

Football is a game that emerged in medieval England. They called it football because you played it on foot, not horseback. Medieval football is vaguely similar to the sports it shares a name with today: two teams fight to control a ball and bring it to the goal on their end of the field. In those days you could carry the ball with your hands, and the field was usually the main street in your town or village. There were tons of regional variations, and over time these variations took on their own characteristics.The town of Rugby had a set of rules people there liked, so those eventually standardized into what we call Rugby. When football came to the US it developed into Gridiron football, or American football. Back in England, the sport was standardized into more or less its modern form, and in the proccess it picked up a nickname: Soccer, from a slang convention applied to "Association football". As I understand it, this nickname wasn't very popular in England- it had a class character, and it was seen as posh, appropriative, and alienating to working class fans and players, who preferred its old name of football. Soccer is the word that caught on when the sport was introduced to the US, because we already had a sport called football. A popular urban legend is that English football fans stopped using the word soccer because it's the word American fans were using, but it's more about that class dimension I mentioned earlier.

In Ireland, one of our national sports is Gaelic Football. People often refer to it as "football" and use the word "soccer" for that sport. So it is not just in the USA. There are many sports referred to as "football" around the world. Soccer and rugby are big in Ireland, so we have several popular forms of football.

Just as a very important aside. The "foot" in football from the original game, not the american one, was not about realted to 'feet' or kicking. The "foot" was that it was played on foot vs being played on a horse, like polo.

Football, was a ball game played on foot, not a game played with your feet. That makes sense now to think that there became a lot of different variations of games called "football" each with their own history.

It's "always" been "soccer" in English-speaking world outside of Britain—and for many people within Britain, too. The British stopped using the term relatively recently, possibly because it became perceived as an Americanism.

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The term "football" was applied to a countless number of medieval ball games played on foot (hence foot + ball). As with other professional sports, these games had very localized or ad hoc rules that were not codified until the 19th century. Notably, in the UK, the Football Association was founded in 1863 and the Rugby Football Union was founded in 1871, which became the two major types of football played in England.

According to the OED, "soccer" as a colloquial name for association football and "rugger" as a nickname for rugby are attested first from 1885 and 1889 respectively:

>1885 Oldhallian Dec. 171 This was pre-eminently the most important ‘Socker’ game played in Oxford this term.
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>1889 Boy's Own Paper 6 Apr. 431/3 In Varsity patois Rugby is yclept ‘Rugger’, while Association has for its synonym ‘Socker’.

As the latter citation suggests, these were primarily nicknames used among the upper classes.

By the 1880s, however, a game called "football" had already evolved elsewhere in the English-speaking world, so when these new games were imported from England, they were known by different names. "Rugby" is distinct enough, but perhaps "association" was too ambiguous, and "soccer" became the name of the game in Australia, Canada, Ireland, South Africa, and the U.S. Some people in New Zealand refer to rugby league as "football" as well.

Two University of Michigan professors, Silke-Maria Weineck and Stefan Szymanski. wrote a whole book on the terminology fight in 2018, It's Football, Not Soccer (And Vice Versa). They point out that the term "soccer" was commonly used in the British press until about 1980—around the time that the game was becoming newly popular in the U.S. Essentially, the term "soccer," invented at Oxford, became coded as an Americanism, and then no self-respecting Brit would want to use it.

If you look at the Google NGram of "soccer" in the British and American corpora over the last 50 years, the divergence comes later than the book, but it is clear and very sharp—"soccer" was in roughly equal use in UK- and US-published books in 1970, but it is a distinctively American term by 2019.

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The term "soccer" is believed to have originated in England in the 19th century as a shortened form of "association football," which was the original name for the sport. In the United States, the term "soccer" has been in use since at least the early 20th century, and it is now the most common term used to refer to the sport in the United States.

In every place, the dominant code is known simply as "football", and other codes are described more specifically.

American football is basically a variant of Rugby football years ago which was called rugger to mark the difference between it and association football again called soccer to identify the game. Americans switched from the naming to American football rather than American rugby or American rugby football to emphasise both the American nature of the sport and it being different from rugby, then the shortened version of American football became football.