It might be good to define what shipping actually is.
Shipping is a term that combines friendship, relationship, and worship together. These are the three escalating means of affection two or more people have for one another, and the basis for shipping.
Shipping was popularized in anime. Unofficial pairings were created for two characters and be paired up in fanon. It has since overtaken every other aspect of any possible fanon out there. Shipping has existed since the beginning of time, when people would gossip about who likes who.
Shipping can sometimes get violent, as two zealous fans defend their respective ships. Shipping can get people in trouble, as one person might be mad at the person making the ship for creating it.
Lots of people will ship themselves with their favorite characters. Fanfiction and fanart are staples for shipping.
Other terms are pairings and couplings. The earliest ship was between Kirk and Spock of Star Trek fame, but the practice wasn't as popular or mainstream until Pokemon came about, in which shipping was rampant and spread like a disease to other anime and eventually to Western culture.