It is when people pick and rant at each other, over inconsequential or just silly things, or what everyone else considers to be such. Obviously, they care about whatever it is, but they tend to just get nasty.....The traders over on Addie's Board do it quite a bit -- if they need to "flame", then they should do it privately. Sometimes people flame in public just to garner support. "Flame" email is when someone sends you nasty, screaming email (we've had a few sellers on ebay do this when we tell them that they are selling a counterfeit). You can also think of this as a verbal attack - many times not warranted by the person it gets directed at.
My daughter introduced me to this term a couple of years ago - apparently there are "flame boards" on the net, where people go just to scream at each other. Wow. I always think "Get a Life".