The referee controls a game of football, implementing the laws of the game (whether the sport is played in the United States or the United Kingdom). Perhaps no other team sport leaves such a wide margin for individual interpretation of the rules as football does.
Knowing what a referee has signalled and understanding why he has done so are essential to your understanding of the game (though it won’t stop you yelling at him). These are some of the most important hand signals you see referees use.