Association Football is the first name given to the sport known as soccer. When the game was first created and rules were codified, there were other types of football being played, so Association Football was the official designation given to the sport. While this term isn't used widely, it is still the technical name of the sport we call Soccer in the United States and Canada.
While soccer is a truly global sport, the version most commonly broadcast on TV and followed by global fans is formally known as Association Football. This is the style of soccer that's played professionally and internationally, and governed by the Federation Internationale de Football Association, or FIFA.
Regions and countries will have their own football associations, such as the Union of European Football Associations (UEFA) or the Confederation of Central, North American and Caribbean Association of Football (CONCACAF). The most notable and popular FIFA tournament is the FIFA World Cup. In the US, professional soccer is largely represented by Major League Soccer or MLS.
Elite players often compete on both a club team, such as Manchester United or the Seattle Sounders, and their national team. A player must have citizenship in that country to compete on a national team. National teams often play international friendlies, or non-competitive matches, to prepare themselves for larger tournaments such as the FIFA Women's World Cup or the EURO Cup for European national teams.