Puppyshipping is the pairing of two fictional characters (mostly anime related) that involves one of the couple being referred to as a dog or puppy or being able to turn into a dog or puppy. It combines the words "Puppy" and "shipping", as in relationship.
Here are some of the most popular puppyshipping parings:
Jounouchi Katsuya (Joey Wheeler in U.S version) and Kaiba Seto - Yu-Gi-Oh!
Inu Yasha and anyone
Remus Lupin and Sirius Black - Harry Potter