|Name Smoosh, Smushname, Fansmush
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A portmanteau (or portmanteau word) is created when two words or names are combined (or "smooshed") to form a new word.
For example, Justin Timberlake/Chris Kirkpatrick becomes Timbertrick, and Clark/Lex becomes Clex, while the pairing of Chloe Sullivan and Lex Luthor, in contrast, was known as Chlex. No particular convention other than euphony regulates how the names are combined.
Well, okay, euphony and hilarity. Some portmanteaus are intentionally funny, such as "Pepperony" for Iron Man pairing Pepper/Tony. Likewise, Minim_Calibre coined the name-squish "Spred" for Spike/Fred on Angel, and dubbed her archive for same "The Buttery," thus demonstrating that name-smooshes are mocked as often as they are found in ordinary use.
In rarer cases, a portmanteau may also include words other than the names of two characters, for example "Wincest" in the Supernatural fandom. The name combines "Winchester" with "incest" to refer to shipping the Winchester brothers together.
The practice almost almost means a romantic or intimate relationship, but not necessarily always! An early example of this is "Chesterbelloc," a phrase coined by George Bernard Shaw in 1918 to describe the combined influence of two writers, G.K. Chesterton and Hilaire Belloc.
Popslash is credited with popularizing the smushname as pairing name. It spread to other fandoms (Buffy, Smallville, Harry Potter, etc.), though only in the age of Tumblr has the smushname taken over completely.
A fan in 2004 said: "Buffy is, as always, a prime suspect I've been seeing "Spuffy" and "Xandrew" and "Spander" for at least 3 years." 
As pet peeve
Fans are divided on the use of smooshed pairing names. Some have fun coming up with them and like the shorthand, while others find them grating. Portmanteaus can lead to silly or unfortunate creations like Spirk and Kock, for Kirk/Spock.
Many other terms in fandom are actually portmanteaus, such as drawble for "drawing drabble", or fanon for "fannish canon", but when a fan says they hate portmanteaus, they generally only mean the infamous portmanteaus-as-pairing-names.
In East Asian Fandoms
The order of the names smushed often indicates who tops/bottoms in the relationship. Often, this is not related to any sexual dynamic; the top is usually the one who is more of a pursuer in the relationship(more active) while the bottom is usually the pursued(taking on a more responsive role). Since the 2010s, many Western fans consider this an arbitrary and outdated practice. Nevertheless, it remains the case with fans in the East, and differing perspectives on its application more often than not cause misunderstandings in communication that may upset either party on social media or fanwork platforms.
In Japanese Fandoms
Ships in Japanese fandoms are also sometimes smooshed, though such combinations generally follow more of a set pattern than in Western fandom. Written Japanese uses morae (syllables) rather than individual letters, and it is common in Japanese to create portmanteaus (or abbreviate words) by reducing longer phrases to three or four syllables. Pairing names follow the same linguistic rules; therefore, Japanese-fandom pairing names are generally four syllables, two each from the beginning of each name. For example, Inui x Kaidou becomes InuKai and Tezuka x Ryouma becomes TezuRyo(u) (the final u is often omitted in romanisation). Unlike in Western fandoms, such pairing names are usually written with each half capitalized, e.g. Sasuke x Naruto is SasuNaru, rather than Sasunaru.
As the majority of English-speaking Japanese-canon fans do not know Japanese, they may use this naming style without fully understanding the rules behind it. Thus the occasional appearance of pairing smooshes more akin to Western styles, that don't conform to the rules of Japanese, such as Tai(chi)/Sora, a popular het pairing in Digimon; this is often written in English-language fandom as Taiora, a combination that is impossible in Japanese.
The order of names in a smoosh for a Japanese fandom (and occasionally Western) can also indicate who tops/bottoms in the relationship, with the top's name being first and the bottom's being second; for example, SasuNaru would have top!Sasuke and bottom!Naruto, while NaruSasu would have top!Naruto and bottom!Sasuke (and SasuNaruSasu or NaruSasuNaru might imply a switch relationship).
In Chinese Fandoms
Smushnames are very common with Chinese pairing names, usually by pairing a distinctive character in the person's name with another character in the other person's name. Which character is chosen is sometimes chosen for which limits ambiguity and sometimes which "sounds nicer" to native language speakers; the order usually depends on which character is the top and which is the bottom.
There is some ambiguity in the pairing names, as a pairing name might simply indicate a close platonic or familial relationship. For example, in QZGS fandom 双叶 (shuang ye - double Ye) is used to tag Ye Qiu & Ye Xiu gen fic as well as incest. Works may also tag 年下(year below) or 年上(year above) tagged; if it is tagged 年上 the top is older, and if it's tagged 年下 the top is younger.
On fannish tumblrs, portmanteaus are widely used, such as #Cherik for Charles/Erik, or #CroWen for Cristina Yang/Owen Hunt; Tumblr does not support the "/" symbol in tags. Posts must be tagged in order to be widely seen, so portmanteaus are more of a necessity. The end result is that smushnames are ubiquitous in fandoms that started (or were rebooted) after Tumblr, but are less common or less dominant in older fandoms. Older fandoms that also have a presence on Tumblr are less likely to have smushnames.
In recent years, gossip magazines and blogs have picked up the trend, perhaps from Soap Opera fandom, and started giving real life celebrity supercouples their own pairing-smoosh names. Some examples would be Bennifer (Ben Affleck and Jennifer Lopez), TomKat (Tom Cruise and Katie Holmes), and Brangelina (Brad Pitt and Angelina Jolie). The supercouple smoosh name trend became so prevalent that Stephen Colbert parodied it on The Colbert Report by citing his favorite celebrity pairing name, "Filliam H. Muffman" (Felicity Huffman and William H. Macy). 
- On Pairing Names, Archived version by cereta (July 7, 2003)
- A Linguist Explains the Grammar of Shipping at The Toast
- "The Fandom Pairing Name: Blends and the Phonology-Orthography Interface" by Cara DiGirolamo, 2012