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Derived from the word "relationship" this term refers to relationships (usually romantic/sexual) fans want to see happen, very common among anime fans.
Shipping is a common among anime fans, an example of an anime with many different shipping couple is Pokémon. The term "Poké-Shipping" refers to the relationship between Ash and Misty while the term "AdvanceShipping" refers to the relationship between Ash and May. Fans will back or "ship" the couple they like and often argue why it is better than another.
Shipping can be between virtually any characters, whether it's a heterosexual relationship or otherwise.