It all depends on the country you happen to be in. The British call it ‘football’, whereas ‘soccer’ is the go-to term used in the USA, Canada, Australia and Ireland.
The word ‘football’ originated in 1863, which was when the English Football Association laid down the rules for this new sport. To set it apart from ‘rugby football’ – already popular at the time – the new sport was called ‘association football’.
And while rugby football eventually became simply ‘rugby’, in England, Scotland and Wales the ‘association’ part was dropped and the name was shortened to just ‘football’.
Things developed a little differently in the USA, though, where a sport originally called ‘gridiron football’ emerged. It incorporated elements of both rugby football and association football. Due to its popularity, it rapidly became known simply as ‘football’.
This meant that when the European football game crossed the pond, a new word had to be found. So it came in handy that students at Oxford had already shortened the term ‘association football’ to ‘assoccer’. And at some point over the years, the ‘as’ at the beginning of the word disappeared completely, never to be seen again.