They were originally a works team -- what we in the U.S. would call a company team, as both the Chicago Bears and Green Bay Packers began -- at the Royal Arsenal (hence the nickname "The Gunners," which led to their fans later being tagged "The Gooners") in Woolwich (the 2nd W is silent), in what was then the county of Kent but is now part of Southeast London.
They called themselves Dial Square Football Club, after the section of the Arsenal where they worked. On Christmas Day, December 25, 1886, they met at the nearby Royal Oak pub, to make things more official, and changed their name to the Royal Arsenal Football Club.
The 1st game under the new name, and the 1st game that could be called a "home game" for them, was on January 8, 1887, at Plumstead Common, 2 miles away, against a nearby club called Erith, and Royal Arsenal won, 6-1.
(The Royal Arsenal at Woolwich. Photo dated 1960.)
In 1893, they turned professional and were admitted to England's Football League, but had to change their name, as a law prohibited any professional team from having "Royal" in its name. (This is not the case in Spain, where there are professional soccer teams named Real Madrid, Real Mallorca, Real Betis, etc.)
So they took on the name of their locality, and were Woolwich Arsenal from then until 1913, when they moved to a new stadium at Highbury in North London, "The Arsenal" until 1919 and "Arsenal Football Club" since, although "The Arsenal" is frequently still used, especially by the club's fans.
They could have changed their name to "Highbury Arsenal," or "Islington Arsenal," even though neither the Borough of Islington nor the Highbury neighborhood had any connection to the Royal Arsenal, aside from the club. As a result, Arsenal are 1 of only 2 professional clubs in England without a locality as part of their official name. The other is Port Vale, located in Stoke-on-Trent, Staffordshire.
Not quite the 1st Arsenal team, but perhaps the earliest known team photo, 1888.
While we sing this song, we'll win the game!
Happy Anniversary, and come on you Gunners! (COYG)