The concept of a Twitard apocalypse is the idea that the Twilight craze will eventually spawn enough insane fangirls that they break out in a full-blown war against all those that dare to not like the books. Of course, this hasn't happened yet, domestic attacks aside, but you never know...
An apocalypse of twitards was suspected to happen when the craze first got so huge, when the movie came out so tons of teenage girls started to oogle over a pale guy staring at them. The theories started to almost seem realistic after attacks from angry fans were reported on people with a less enthusiastic opinion on Twilight. But by the third movie, the hype seemed to be dying down, which is why most antis no longer fear this possibility.
Unfortunately, there is another scary idea on why this might ever happen. Because Breaking Dawn already split the fandom: the most rabid fans still clinging to it, and the ones who hated it and sometimes turned into antis afterwards, there is a possibility that when the movie comes out, the former will become more rabid and defend their new movies to the death. There may also be a chance that when everything seems like it's over, guess what happens? Midnight Sun is published, and all the fangirls respawn like zombies back from the dead.
People used to think that the Twisease would be what takes over, causing a plague turning all the people in the world into insane Twifreaks except for the few sane people of earth who hide underground in a resistance base until the hell seemingly ends.
The worst case scenario. If this happens, we will have to fight to the death. It will be a never-ending, bloody struggle for the sake of good literature.
Twitards fight using whatever makeshift weapons they can find, traditionally based around the "symbolism" of Twilight. So, imagine chess pieces as weapons, apples to pelt at people, and ribbons and flowers used for gagging and certain odd forms of torture. Others just beat people up using what odds and ends they have.
If you are ever to enter a war on Twilight, never bring people who hold a neutral opinion on the books/movies or haven't read/watched them. Because rabid fangirls will hurt anyone who even says that they have no opinion on Twilight, so innocents must not be jeapordized. Also, do not mistake sane fans for Twitards, because they would not hurt you for your opinion. This is our battle, the antis against the twitards. If it ever comes, we will fight for our opinions, and possibly and hopefully redeem the fangirls with logic, so peace will reign at last.
If you want to know how to survive a twitard apocalypse, see Twitard Survival Guide.