A heresy is a belief that doesn't agree with the official tenets of a particular religion; heresy is the maintaining of such contrary beliefs.
Heresy can be used figuratively: to disagree with the school committee's decisions is considered pure heresy by the faculty. You'll often come across this word in a religious context — the Latin root haeresis, "school of thought or sect," was used by Christian writers to mean "unorthodox sect or belief." This comes from the Greek hairesis, "a taking or choosing," from hairein, "to choose." A person who chooses to believe in heresies is called a heretic.