"And what fastens attention [...] like any passage betraying affection between two parties? [...] We see them exchange a glance or betray a deep emotion, and we are no longer strangers. We understand them and take the warmest interest in the development of the romance. All mankind love a lover."—Ralph Waldo Emerson
Rooting for fictional romance to happen.
The term "Shipping"—which ostensibly derives from "Relationship"—was originally coined by fans of The X-Files, who were divided between "relationshippers" pushing for romance and "noromos" who would rather have No Hugging and No Kissing. The phenomenon itself, however, was ubiquitous in practically every fandom long before. The source of the term's popularity is shrouded in myth, but Geek Mythology has it that you can blame people in the Pokémon anime's fandom who root for Jessie and James (of Team Rocket) to get together and decided to call themselves "Rocketshippers". This got the term to catch on in that fandom, then people brought it with them to other fandoms and this is why as of today The Other Wiki needs a tiny disambiguation note at the top of its article on maritime delivery of goods.
Of all the obsessions that universally afflict fandom, Shipping is by far the most persistent, widespread and prone to be Serious Business. It knows no boundaries of age, demographic and gender. This might raise an eyebrow or two on first inspection, but honestly, shipping is just a consequence of plain old human nature: we are wired to seek a romantic partner in such a powerful, fundamental way that we even get a considerable kick out of doing it by proxy—and fictional characters are plenty, easy to relate to, often in want of someone to make out with, usually get their happy ending and nearly never fall out of love. We wish we were them. We are drawn to reimagining ourselves in their place like moths to the flame.
Somehow, this leads to vitriolic forum threads with a hundred times the activity of all the other threads in the forum combined, titled something like "will Bob get together with Alice or Eve? Round 997!"  Which in hindsight is understandable, given that when a lot of fans are projecting themselves into one character, it's bound to get very crowded in there. Really, put just two fans in a room and give them five minutes or so, and they'll get into an argument to the death about some incredibly fundamental fandom issue; having hundreds of fans in the same internet is just asking for trouble. And when you ask for trouble what you get is, in the words of J. K. Rowling, scary and vehement cyber gang warfare.
Shipping is all about the anticipation. Paragraphs and essays and counter-essays weighing megabytes at dozens of pages will be written about who will get together, who should get together, and what the disciplines of political science and feminism and probability theory have to say about the issue (the above is Not Hyperbole). No ecstatic shipper has ever written a gigantic dissertation titled "Hurray!! Alice and Bob! FINALLY!!" or any fandom equivalent. People will argue endlessly about the romantic future of nearly any given ensemble, but if that point should actually be resolved, the discussions will basically go through a round of ranting and gloating and then unceremoniously run out of steam.
That's probably because anticipation is something that's easy to feel you're a part of, even if the anticipation is for something fictional. Real life romance, for all its shortcomings, actually happens for us Real Life people: We move on from looking forward to something great to experiencing something great (or at least we can hope). Fictional romance not so. Actually being in a romantic relationship and getting to watch a fictional romantic relationship are very different things, much more different than looking forward to each of those respectively. The contrast is jarring—you were a part of this great love story, and now suddenly you're not. Cue disillusionment. Of course, the near-universal reaction is to move on to the next fandom, making shipping suspiciously similar to an addiction.
Canon and authorial intent do not dictate people's shipping preference. You'd be hard-pressed to find anything that honestly puts any sort of restraint on shipping preference. Characters may be shipped despite being still in grade school, of the wrong sexuality, siblings or twins, Just Friends, mortal enemies or just generally the bane of each other's existence, separated by an age gap of decades or centuries, not of the same narrative continuum, part of a story where romance just isn't an issue, inanimate objects, nigh total strangers, considered as a possible couple at all only because they're both left single after you're done pairing everybody else, extremely implausible as a couple by design or even outright denied to ever possibly get together by Word of God. Even when the source material goes as far as to have an Official Couple, sometimes fumbling execution or Values Dissonance will drive fans towards emotional investments diametric to those the author intended.
Shippers have a reputation of insane devotion to their One True Pairing and of interpreting the tiniest, most ambiguous details as evidence. That much is clear by the prevalence of Shipping Wars in any fandom discussion. In some extreme cases they will freely admit to actively rooting for sympathetic characters to die just to get them out of the way, or worse, they'll come to the conclusion that since a character is in the way, they are by definition not sympathetic. On the bright side, you can expect them to be friendly at least towards their natural allies and have some limit of how severely they can be starved for validation before they Abandon Shipping. That is, if they ever expected to be validated in the first place.
Frequently authors know full well how loaded the subject is and tease the audience of actual and potential shippers, sometimes to the extreme of letting the characters themselves join in on the shipping fun. It goes without saying that not all ongoing tease storylines endure and graduate to Relationship Upgrade; and then a majority of the ones that do don't until near the very end. This is usually not done to frustrate fans (or at least, not for this sole purpose), but rather because the writers are also aware of romance being a very powerful device that should be handled with great caution, lest it swallow the storyline whole or reach a resolution past which people stop caring. For the audience, being thus denied leads to frustration, and frustration leads to the dark side, that is, writing Wish Fulfillment fan fiction as a form of emotional venting.
There's a whole nomenclature dedicated to Quick, Easy and Idiosyncratic Ship Naming, often varying from fandom to fandom. The most basic tool of communication here is the slash—if you wanted Alice and Bob to get together you could always say you shipped Alice/Bob. However, for most fandoms that's just not exotic enough. They will not be content with anything less than a short, sweet and catchy brand name—the more Incredibly Lame the Pun, the better (Harry Potter fandom actually named ships the "HMS this-and-that"). Shipping culture has also imported the Portmanteau Couple Name from Japanese Anime fandom; apart from its infamous usage in the gossip industry ("Brangelina", "Bennifer", "TomKat") you can find people online declaring themselves fans of "Pepperony", "Wuffara", "NaruHina", "Sheelos" and "Jam". Yes, Jam. Needless to say, if some series has Theme Naming shippers will exploit it Just for Pun—In Real Life, 3/4 is just a rational number greater than one half and lesser than one. Not so in Codename: Kids Next Door fandom.
As far as omnipresent forces of nature go, shipping is up there with Death and Taxes; an "Example Section" would just be a list of works popular enough to have a member of their fandom happen to come across this page. Suffice to say that on any show, "Alice/Bob is popular and so is Eve/Mallory, though there is lots of vitriol between supporters of Gum/Popsicle and Twix/Popsicle", up to minor variations. For an example with some unique flavor to it, you might want to look into Shipping Tropes.
- contrary to expectations one might reasonably have involving Yaoi-worshipping, fan fiction writing, Chick Flick-addicted, squealing teeny boppers
- For all we know, this very well may have actually been a thread title in a Cryptography forum.
- Or, for that matter, the Gundam Wing fandom. Take your pick.