The term "shipper" comes from supporting a ship. To ship something means a person wants two characters to get together and/or shows support for two characters already together. The term "ship" came from the X-Files fandom, when fanfics were written about Mulder and Scully. The fans then called themselves shippers. It quickly spread and is now the title a person gives themself if they believe two charcters should or will be together (The characters can be from anything: Books, Movies, Television, Video Games, and even Actors/Actresses). It is not limited to the couple actually happening, a person can ship something just because they enjoy the possibility of them getting together or even just because they think they would look good together.
I am a Harry/Hermione shipper in the Harry Potter fandom.