The reminders posted by @marichelAlvarez inspired me to use Teen Wolf as the examples for this explanation of shipping as a fandom phenomenon. Fanlore defines shipping as "the act of supporting or wishing for a particular romantic relationship". The term originally surfaced in the X-Files fan community, but it has grown to encompass any fandom's pairings. So if you were rooting for Elizabeth and Mr. Darcy when you read Pride and Prejudice, you're a shipper. Congratulations!
Some fans are multishippers- they're dedicated to more than one couple. Other fans are invested in their OTP (one true pairing) and nothing else. Both are valid forms of shipping. The Teen Wolf fandom is a rich community with a variety if ships and sub communities. It's an excellent case study of fandom dynamics. (Beware: there will be SPOILERS).
These are relationships that exist within the canon. They're not solely subtextual, and don't thrive on speculation, because they're written into the main story. In Teen Wolf, Allison and Scott dated in season 1, and broke up in season 2. They both started dating other characters. These ships are canon because their stories are an integral part of the plot of the show. Some fans get very invested in canon ships. This makes sense- lots of stories are driven by romantic relationships, and becoming invested in them is part of the natural viewer experience. However, these ships don't always have a happy ending. Allison (sob!) died last season. Canon ships don't necessarily rely on the original media, but they are often inspired by it.
Fanon is the opposite of canon: it exists outside of the original media, exclusively as part of the fan community. Some fanon ships rely on subtext and speculation, but others are derived solely from personal interest. The Sterek community (Stiles/Derek) is by far the most popular. With over 34,000 fanworks on Archive of Our own (a popular fanfiction website), it's clear that a ship does not have to be canon to be embraced by fans. Some fans create these ships so that they can experience the representation that the show doesn't deliver. That's why a lot of fanworks are LGBT. Even though Teen Wolf has a somewhat progressive track record, featuring a few canonically gay or bisexual characters, none of them have been as prominent as the characters presented as heterosexual.
Within fan communities, there are often niches that ship lesser-known characters. These rarepairs can be canonical or fanon, and are obscure because they're less popular among fans. What they lack in popularity, they often make up for with passion. Rarepair shippers create challenges or exchanges to communicate with each other and create more content for their ship.
NOTPs, Wrongships, and Enemyslash
Some fans explore the dynamic between two opposing characters through shipping. This goes a little further than the frenemies to friends trope you might be familiar with. They pair an antagonist or enemy with the hero, and see how that changes the relationship. Some fans do this to clarify or elanorate on something the original media didn't expand on (like the manipulative behavior Kate Argent used to trick Derek Hale). Others do it because they're fascinated by the characters.
Shipping is a way for fans to share their passion and find like-minded fan communities. It's not a new phenomenon, and it's actually been added to the OED. Shipping is a valid part of the fan experience, and the results are often creative and encourage fan engagement with the original media. While it's awesome to see fans exchanging fanworks and metas, it's less fun when fans engage in flame wars over their ships. The One Direction fandom has been plagued by this problem, and Teen Wolf fans aren't immune. The best way to be a responsible shipper is to keep your famworks and discussion in the appropriate communities or tags.