Before I begin this post proper, I should explain what forking means in the technical and practical sense.
Forking is a tech term for taking software (almost always open-source, though some proprietary software allows for this under certain legal and technical conditions) and making your version based off an existing branch of the software's development, which may or may not follow the path of the original software while still retaining the base code behind the original.
Tv Tropes uses a customized fork of PmWiki, which is a freely modifiable wiki engine, this is a technical fork made to suit their needs.
However, forking is also a social tool, as forking can be used for websites and website concepts. 8chan is a fork of the 4chan concept. Voat is a fork of the concept behind Reddit, Slimgur is a fork of the idea of Imgur and my own site, All The Tropes, is a fork of TV Tropes.
When websites fork, it's usually due to ideological splits with the site they forked from, and in the case of all the sites mentioned above, all forked from the websites behind their core concepts due to ideological opposition to censorship and those imposing it, and thus went to the forks where their wishes would be respected as a community, an idea I support wholeheartedly because censorship based on ideology is a slippery slope that does no one any favors in the long term.
Forks of websites may not always be identical in terms of function or technical structure, often for either legal or practical reasons, though most of the sites I mentioned that are forks are forks for practical reasons, though ATT is a mix of both since PmWiki was discarded as unsuitable for our goals and needs and because certain technical aspects of TV Tropes fall under potential copyright concerns I felt were best avoided by using a different wiki engine. Our site content is also based on the last legally viable point when the site content could be forked under acceptable licensing terms for redistribution.
Forking is often either derided or attacked by the source websites, mostly due to fear of revenue and userbase losses. ATT was attacked by Fast Eddie for these reasons, though the current administration of ATT and TVT have, after initial teething problems, settled into a period of benign indifference to each other so long as certain legal conditions are honored, such as not plagiarizing each other's content, which we have mutally agreed to respect and enforce. We have also promised each other to cooperate when dealing with vandals and other miscreants, and I personally hunted down information on someone who made a phished clone of TV Tropes and wrote a blog post on how to spot phished webpages because of my intense loathing of cyber crimes. TV Tropes has honorably rescinded their claim to retroactively own all edits made to their site and that users surrender their rights to redistribute, a fact we still note with gratitude.
It bears mentioning, though, even though I'm now a Gamergater (who are oft accused by their detractors as harrassers and criminals), that I firmly fall on the side of keeping a fork within the bounds of the law, and that all activities on that fork should be as ethical and aboveboard as possible, not only for obvious legal reasons, but because if a fork really broke from its source due to a perceived corruption of their founding principles, then it behooves the community of the fork to act more honorable than the place they left, else they be revealed as hypocrites.