These articles or e-mail messages ridicule another or another's ideas, often in a stinging, blunt manner. Flames are a tradition on the Internet, and something to expect if you make a netiquette blunder. Do not necessarily take them literally, but look for the suggestions presented. The tone is generally very negative, but the point is usually clear.
The act of flaming someone is the act of responding in a highly critical sarcastic or ridiculing manner. The name presumably derives from "to shoot down in flames". Anyone who posts an offensive article can expect to get flamed, probably by more than one person. Vicious arguments between two or three sides often become know as flame wars.
Flames are the opposite of helpful feedback. It means sending an email to an author to tell them you disliked the story you just read. Common examples are people who can't stand slash, yet read a slash story in its entirety then email the author to tell them they hate slash. Well, here's a thought, if you know you won't like it, don't read it. Remember there is a difference between saying 'you may not know it but you spelled bestiality wrong.' and saying 'if you can't spell it you shouldn't be doing it, dumb ass'. While I realize the latter is more tempting be polite, authors have memories like really well superglued traps, they won't forget your flame and will probably even send back a snarky response if you send them one.