Shipping refers to fan theories about (usually fictional) characters being in a romantic or sexual relationship.

The term comes from a sect of X-Files fans who called themselves "Relationshippers" because they believed the show was building up a romantic relationship between the main characters of Mulder and Scully, but the concept dates at least back to the Kirk/Spock subculture within the Star Trek fandom, and possibly even further depending on how broadly you define shipping.