First, are we referring to American football, soccer (what most of the world calls "football"), or another varient such as Irish football or rugby? Arbitrary, we will say this question is asking about American football. See the linked question to the right for when soccer was invented.
Next, what does it mean for a sport to be invented? American football evolved from other games, specifically soccer and rugby. Again, refer to other questions for answers as when those sports were invented.
Finally, are we really asking when the NFL -- the professional American football league -- was invented? Once again, see the related question for that answer.
Now the question is more narrow: when did the game that Americans know as football take its current form?
The answer is that football was invented in the late 1800's, the late 19th century.
There is some dispute about the exact year. Do you have evidence for which year is correct? Here are the answers that have been given:
1886 is when Walter camp invented American football from the English rugby and soccer game.
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The best answer is indeed 1880.
In 1869, Rutgers and Princeton began the tradition of intercollegiate fall team sports by playing each other in variations of soccer. Throughout the early 1870s, Yale, Princeton and Columbia continued playing soccer variants while Harvard, Tufts, and McGill played variations of rugby.
In 1875, in order to challenge one another, Harvard and Yale agreed to "concessionary rules" which combined elements of both sports, but notably included running with the ball and tackling. Though Harvard won the match, the Yale players and the Princeton team which had attended as spectators were both converted to the running and tackling game. This is the point at which soccer interest in the U.S. became relegated to a secondary status.
Walter Camp played the rugby version of the game at Yale in the late 1870s, but was frustrated that the randomness of the rugby scrum often prevented clear-cut offensive strategies and plays. After the 1879 season, he convinced the Intercollegiate Football Association to adopt for 1880 a drastic change in which clear possession of the ball by one of the teams would be acknowledged, the circular rugby scrum would be straightened out into a "line of scrimmage," and the game would be structured into individual plays which would begin with the snap of the ball by the center and end with the tackling of the ballcarrier. These are the essential elements of American football, and it may be reasonably concluded that whichever college game occurred earliest in the 1880 season was the "first" true game of American football. Since 19th-century intercollegiate football was arranged not by institutional Athletic Directors but by groups of undergraduates, it's difficult to ascertain with certainty the earliest game date or game time of 1880 matches plated under the new rules, but a strong candidate for the first game was Princeton's 6-0 victory over Stevens Tech.
While Camp was certainly a visionary, he had overlooked one key element for the new version of the game. It was tradition among the northeastern schools at the time that the winner in a given year would receive a championship banner, and that in the event of no clear-cut winner, the previous year's champion would retain the banner. The teams were more interested in possessing the championship banner than in winning an individual game, so in both 1880 and 1881, Princeton - the 1879 champions - simply held the ball against Yale, running safely into the middle of the line for no gain rather than attempting to score. In this way, they managed to ensure that both games ended in 0-0 ties and they retained their 1879 championship. At the close of the 1881 season, Camp proposed and the Association concurred that from that point on, teams would have to gain five yards (or lose ten!) in three downs in order to maintain possession of the ball. One could argue that 1882 marked the first emergence of the down-and-distance game, though I'd suggest that down-and-distance was simply a refinement of the new game of clearly-defined plays which debuted in 1880.
Clearly there is quite a bit of controversy on this topic so I will give the range of answers right here -> 1866-1895
Answer
Gridiron football (not futbol or soccer) was actually born in Canada by British soldiers against civilians in a game in Toronto on November 9, 1861. It has been modified since then.
1920. The name was changed to National Football League in 1922.
The American Football League first played games in 1960.
The National Football League, at the time called the American Professional Football Association, was founded in 1920.
Average of 8 deaths a year in American football.
1987
1920. The name was changed to National Football League in 1922.
Rugby
The American Football League first played games in 1960.
The National football league started in 1920.
The National Football League, at the time called the American Professional Football Association, was founded in 1920.
American football started in the year 19th century. Some people refer football as soccer.
The Chargers were a charter team of the American Football League when it began play in 1960. That year they played in Los Angeles and moved to San Diego in 1961.
Average of 8 deaths a year in American football.
1968
In 1920.
1987
Canton, Ohio. Where the Pro Football Hall of Fame is.