A very public and off-topic conversation, often between only 2 people, in a public forum such as a news group or email list. Reason and logic seem to go out the door, and "ad hominem" attacks are plentiful. GodwinsLaw is sometimes (unwittingly) breached.
One famous flame war was (primarily) between LinusTorvalds and AndrewTanenbaum (a well known cs professor who's written many books about os design, among other topics, as well as writing much of Minix). The first posting is from Tanenbaum, and has the subject "LINUX is obsolete".
It can be found in places on the web - one such place is http://www.oreilly.com/catalog/opensources/book/appa.html
Here is a brief quote from Linus in his first reply. See if you can spot any hostile language:
A FlameWar is defined by the manner of discussion, unlike a ReligiousWar, which is defined by the topic of conversation, and these often overlap.
See also ReligiousWar, GodwinsLaw
4 pages link to FlameWar: