Purported movement to bring down the Ministry of Magic using Dark Magic and gum disease.
Luna Lovegood believed the Rotfang Conspiracy to be a secret movement to bring down the Ministry of Magic from within using a combination of Dark Magic and gum disease (HBP15). Aurors are purported to be part of the conspiracy, to the point that Luna advised Harry against becoming one during Slughorn's Christmas party. Professor Trelawney seemed to be sincerely interested in the topic and continued talking with Luna about it after Harry goes to follow Snape and Draco (HBP15).
Commentary
Etymology
"Rotfang" is likely derived from the Conspiracy's use of gum disease as a weapon: 'rot' + 'fang'.
Notes
There is no evidence outside Luna's statement that the Rotfang Conspiracy existed.
Interestingly, Voldemort tells Bellatrix they will "cut away the canker that infects us" (DH1). While he is referring to those not of the "true blood," it is interesting that he uses a reference to a sore at the base of the gums (by one interpretation) and proceeds to overthrow the Ministry from within using Dark magic and Pius Thicknesse, who was the Head of the Magical Law Enforcement Office that oversaw Auror operations.
Additionally, when Cornelius Fudge and Rufus Scrimgeour meet with the Muggle Prime Minister (HBP1), Fudge at one point attempts to smile but instead looks as though he has a bad toothache. While it is certainly a stretch to attribute this to the Rotfang Conspiracy, it is an interesting tidbit and connection, especially after Fudge had just been ousted as Minister of Magic.