There is something about the anonymous nature of the Internet that makes people behave in ways that they wouldn't when they're face-to-face with other people. What starts as a minor disagreement like "how to ship two minor characters," or "what constitutes Canon or fanon," escalates until the users are insulting each other, flinging expletives and profanities and shouting at the top of their virtual lungs. Inevitably, someone will be compared to Hitler. Also inevitably, the Internet Tough Guy will get involved, resulting in legal threats (anything from DMCA takedowns and people trying to drag the legal staff of people or businesses involved into the flame war to often hilariously misspelled accusations of "defamation" and even possibly an actual Frivolous Lawsuit), and sometimes even actual threats of violence or Real Life harassment.
After a time, the flames may abate, but usually more because of exhaustion than anyone actually getting a point across. Once everyone has recovered their stamina, all it takes is one innocent (or not-so-innocent) person bringing the subject up to start things all over again. In even worse cases, people insist on taking the flame war or their grievances from old flame wars to the people once involved elsewhere, or seek to get uninvolved people involved and taking a side, which is why This Very Wiki and some other sites have strict rules against importing or exporting drama.
In fandom related communities, a lot of the flame wars are the result of a Broken Base, Fandom Rivalry, Fandom Heresy, or other Fan Dumb behavior — which, in turn, is the result of people regarding the fandom as Serious Business. In a way, it can be even more bothersome than dealing with trolls who come onto the community to bash the entire fandom. Sometimes a fandom Flame War can reach such levels that it becomes a Holy War, a massive, prolonged battle of insults. The Console Wars and PC vs. Mac debate are good examples of these prolonged, though rare, wars. And God help you if you actually start discussing politics or religion.
See Internet Backdraft for when a Na´ve Newcomer unintentionally sparks a flame war and Flame Bait for a list of tropes that should never ever be mentioned on this very wiki (or anywhere else, for that matter). See also Misplaced Nationalism for a specific form of flame war.
For a literal flame war, see Kill It with Fire.
No examples, please. There will always be something that can cause this.
Anyone got popcorn?