Merriam-Webster, the Oxford dictionary and other sources define "fan" as a shortened version of the word fanatic. Fanatic itself, introduced into English around 1550, means "marked by excessive enthusiasm and often intense uncritical devotion". It comes from the Modern Latin fanaticus, meaning "insanely but divinely inspired". The word originally pertained to a temple or sacred place [Latin fanum, poetic English fane]. The modern sense of "extremely zealous" dates from around 1647; the use of fanatic as a noun dates from 1650. However, the term "fancy" for an intense liking of something (a usage attested by 1545), while being of a different etymology, coincidentally carries a less intense but somewhat similar connotation to "fanatic". Use of "the fancy" to mean avid sports enthusiasts emerged as an Americanism in the mid-19th C. The Dickson Baseball Dictionary cites William Henry Nugent's work asserting that it was derived from the fancy, a term referring to the fans of a specific hobby or sport from the early 18th century to the 19th, especially to the followers of boxing. According to that theory, it was originally shortened to fance then just to the homonym fans. The Great American Baseball Scrapbook attributes the term to Chris Von der Ahe, owner of the Saint Louis Brown Stockings in 1882. Von der Ahe sold tickets for 25 cents, hoping the many patrons would purchase his beer; the low ticket price helped him lead the stats in attendance. He called the fanatics filling his stands "fans". 
Supporter is a synonym to "fan" that predates the latter term and is still commonly used in British English, especially to denote fans of sports teams. However, the term "fan" has become popular throughout the English-speaking world, including the United Kingdom. The term supporter is also used in a political sense in the United States, to a fan of a politician, a political party, and a controversial issue.