Shipping, initially derived from the word relationship, is the desire by fans for two or more people, either real-life celebrities or fictional characters, to be in a relationship, romantic or otherwise.
Shipping can involve virtually any kind of relationship: from the well-known and established, to the ambiguous or those undergoing development, and even to the highly improbable or blatantly impossible.
Shipping often takes the form of creative works on the internet, including fanfiction and fanart. A 'ship' refers to the relationship supported, while 'shipping' refers to the phenomenon.
The pairings of characters are typically given names—usually portmanteau of the characters names. For instance, Ronald Weasley and Hermione Granger combines the "Ro" from Ronald and "mione" from Hermione to give Romione. While other fandoms, such as Once Upon A Time choose ship names that fit the character's roles and personality. For example, 'Outlaw Queen' is the ship between an evil 'queen' and an archer.
The term originates from the X-Files fandom. The relationship between Mulder and Skully began gaining interest. Some fans occasionally described themselves as relationships or shortened as 'shippers. The use of the shortened term began around 1996. The earliest use of shippers is used in an X-Files form, alt.tv.x-files.creative. At this time, it was shortened to R'shippers. 
The term later develops into shippers. Though for a period of time, shippers usually referred to fans who are interested in heterosexual relationships. While, 'Slashers' referred to people who enjoyed gay relationships in fanworks.
A pairing of characters or real-life people, where one side is male and the other female. The word "Het" is derived from the word, "Heterosexual".
NaruHina – Naruto Uzumaki and Hinata Hyūga (Naruto)
Fresley – Wesley Wyndam-Pryce & Fred Burkle (Angel)
Bandits – Brenda Walsh and Dylan McKay (Beverly Hills, 90210)
Jafael – Jane Villanueva and Rafael Solano (Jane the Virgin)
Slash refers to relationships between two male characters. Often the characters are portrayed in fanfiction or fanart as being homosexual or bisexual. Occasionally, Slash is also used to describe two female characters. Though, Femslash became more popular to describe these instances. In anime fandoms, the word Yaoi is usually used instead of Slash.
Drarry – Harry Potter and Draco Malfoy (Harry Potter)
Destiel – Dean Winchester and Castiel (Supernatural)
ColdAtom – Leonard Snart and Ray Palmer (Legends of Tomorrow)
Femslash, derived from 'female slash' refers to relationships between two female characters. Also occasionally called F/F, Femmeslash, Girlslash, Ladyslash or Saffic. Often the characters are portrayed in fanfiction or fanart as being homosexual or bisexual. Similar to Yaoi, in anime fandoms, Yuri describes femslash pairings.
Korrasami – Korra and Asami Sato (Legend of Korra)
Wuffy – Willow Rosenberg and Buffy Summers (Buffy the Vampire Slayer)
Also called Threesome, Foursome, Moresome or OT3 ships. Poly, short for polyamorous is a relationship between three or more characters. Often, Polyship is used to describe relationships with a deeper and long-lasting relationship. However, the term threesome in fandom is also used to describe both sexual and romantic relationships between three or more people.
Wincestiel – Dean Winchester, Sam Winchester and Castiel (Supernatural)
Pepperstony – Steve Rogers, Pepper Potts and Tony Stark (Marvel)
Non-binary ships are used to describe relationships including one or more non-binary individuals, or ships involving characters whose true gender is unknown, or characters who fit into a fictional gender or sex. Though non-binary individuals are shipped, it is rare to see the pairing described as a non-binary ship. Alternatively, characters who are transgender can sometimes fit into the 'Het', 'Slash' or 'Femslash' categories. In gender, Non-binary is an umbrella term to describe any person whose gender is not male nor female. Some common nonbinary gender identities include: agender, bigender, genderfluid, androgyne and neutrois. It is also common, in fanfictions for writers to change the gender of a character in an Alternate Universe story.
Lapidot – Lapis Lazuli and Peridot (Steven Universe)
The beautiful, timeless, tireless loyalty and love that is friendship.
Grovercy – Grover Underwood & Percy Jackson (Percy Jackson and the Olympians)
In many fandoms, some fans jokily ship a character with an inanimate object, usually food. They are viewed as Crack!Ships, by most fans. The word, Cargo Ship was coined to describe these ships. A character intense love of an inanimate object tends to cause fans to humorously joke about the love between them.
Timeship – Rip Hunter & Gideon, the Waverider's A.I. (Legends of Tomorrow)
Drapple – Draco Malfoy and an Apple (Harry Potter)
Harustache – Haru and his mustache (Avatar: The Last Airbender)