For most everyone in the world, "shipping" is a service provided by Amazon that you always feel that you pay too much for. But for those in the fandom communities, "shipping" is the act of making two or more characters (be it film, literature, comics, tv, etc.) hook up with each other romantically. It's the backbone of most fan-fiction and fan-made drawings and it is as vital to sites like Tumblr and DeviantArt as hydrogen is to water.
Shipping is especially prevalent in media consumed by young adults such as Avatar: The Last Airbender and Harry Potter: The Last Book Most People Have Read. That's why the battles over which cartoon characters should be rubbing their decidedly imaginary sexual parts together can feel a little cringey as evidenced by this thread below:
It's also why (as this Reddit thread unearthed) the series creators of Avatar decided to poke a little fun at the shippers and create this video:
It's an animated short that uses the fan-made art of shipped couples within the series and then plays out the scenario of their relationship. Sure, Katara and Zuko might fit well together in your fantasies, much in the same way that you and Justin Bieber would have been perfect together had he only gotten the chance to know you. But those relationship fantasies rarely factor in Justin Bieber's struggles with addiction or Prince Zuko's mental anguish over his TARNISHED HONOR!
This video seems meant to be fun and lighthearted, but some members within the fandom probably don't see it that way. Shipping wars can be intense as participants will claw on to their favorite ship like it's a literal sinking boat. Here's a tip if you ever want to start a fight on the Internet. Just tweet out something about how you hate or love Kylo Ren and Rey being together and you'll get something like this:
There are holy wars that are more civil and also, apparently, a lot less gross:
All of the pairings this person posted are related to each other or interspecies or carry some other problematic dynamic that goes well beyond taboo. If they're pairing siblings together from a Marvel franchise then I can't even imagine what their Game of Thrones shipping looks like.
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Top Image: Nickelodeon Animation Studios