An Opiate is an alkaloid that is found in the Poppy plant, Papaver Somniferum, and in Opium. The main Opiates are: Morphine, Codeine, Thebaine and Papaverine.
Opioids are synthetic or semi-synthetic derivatives of the Opiates. Examples are Diacetylmorphine (Heroin) and Oxycodone.