Short answers, pulled from the story.
Football has been part of every Summer Olympics as a men's sport except 1896 and 1932. Tournaments were played at the 1900 and 1904 Games, though those were contested by clubs and scratch teams, and a proper international tournament arrived at the London Games of 1908.
Since 1992, male competitors have been required to be under 23 years old, with up to three over-age players allowed per squad since 1996. FIFA imposed these restrictions to keep the Olympic men's tournament from rivalling the World Cup.
Women's football joined the official Olympic program at the Atlanta 1996 Games. Unlike the men's tournament, it is a full senior-level international competition, second in prestige only to the FIFA Women's World Cup.
FIFA dropped football from the 1932 Los Angeles Games to promote its new World Cup tournament. Jules Rimet had proposed a professional World Championship of Football in 1929.
The Soviet Union and its satellite states dominated every Olympic football tournament from 1948 to 1980. Of the 28 medals awarded in that span, Eastern Europe won 23, with state-sponsored athletes who retained amateur status facing young Western amateurs.
In 1950 the English FA abolished the distinction between amateur and professional football and ceased entering the Olympics. The four Home Nations feared a united British team would set a precedent leading FIFA to question their separate status in other competitions.