BRAND NAMES: none. DRUG CLASS AND MECHANISM: Codeine is a narcotic pain-reliever and cough suppressant similar to morphine and hydrocodone.Moreover, a small amount of codeine is converted to morphine in the body.RAND NAMES: none. DRUG CLASS AND MECHANISM: Codeine is a narcotic pain-reliever and cough suppressant similar to morphine and hydrocodone. Moreover, a small amount of codeine is converted to morphine in the body.