Shipping (カップリング Coupling) is the short term of relationshipping. It is a belief that two or more characters are involved in a romantic relationship, or have feelings that could lead to one. It is usually a suffix attached to a word that relates to the two characters. For example: RocketShipping is the shipping of Jessie and James from Pokémon, because the two are both from Team Rocket.
A person who supports a shipping or would like to see them as a couple, would be a shipper. There is also "ship" which means the pairing itself, or "shippy" which is a term used as an adjective. For example, if Inuyasha and Kagome from Inuyasha kissed, it would considered to be a shippy moment.
As a basic rundown...
Noun Ship (pairing). E.g "I support that ship."
Adjective Shippy (supports a ship). E.g "That was a shippy scene." This basically means "That scene makes them look as if they have feelings for eachother."
Verb Shipping (Supporting a pairing). E.g "I like RocketShipping." Basically means "I like supporting this couple."
Person Shipper, someone who supports a ship.
The term "shipping" originated from an X-Files fan community. During that time, a fan was a shipper of the two main characters of the television program and began to use the term. The term was eventually crossed over to a Pokémon community, a Team Rocket community to be exact. From there the first Pokémon shipping was created, RocketShipping. The term eventually floated toward other Pokémon communities.
The shipping term isn't used much among most other fan bases, but it has been present in Dragonball Z and Digimon fan bases. Although it has been seen being used in various other fan bases besides that, most people however do not give ships special names. Shipping is more popular in Pokémon fan bases, where the idea of shipping first grew big. There is over one thousand Pokémon pokemon shippings and more are still being created.