A Facebook "fan" is a user who "likes" a particular page. If a user chooses to "like" a page, they are then able to get updates from that page's administrator through status updates, link posts, and event invitations. Fans display lists of pages they have "liked" on their profile, under the "interests" section of their information tab. In April 2010, Facebook changed the terminology from "becoming a fan" of a page to "liking" a page.
Morin Oluwole explains how to become a fan of something on Facebook.
The Huffington Post discusses why Facebook replaced the "become a fan" button with a "like" button.