What Does Facebook Fan Mean?
A Facebook fan is a user who likes a particular Facebook page. Users who Like a page are able to receive updates from that page’s administrator through status updates, posted content and event invitations. A list of pages a fan has liked will appear on his or her profile page.
The term the fan was replaced by Like in 2010. Now fans Like a page and this is shown on their profile rather than that they are a fan.
Techopedia Explains Facebook Fan
In 2010, Facebook explained the change in terminology from "fan" to "like":
"To improve your experience and promote consistency across the site, we’ve changed the language for Pages from Fan to Like. We believe this change offers you a more light-weight and standard way to connect with people, things and topics in which you are interested."
Liking a page is not the same as liking a piece of connect because this action connects the users to that page, allowing them to receive updates from the page’s administrator. "Like" was chosen over "fan" because it is a more inclusive term that encompasses the wide range of Facebook pages, many of which extend beyond brands and bands; on Facebook, people can "like" just about anything, including sleeping, pickles and the immune system.