Cold process soapmaking method entails mixing lye water with a fat or oil (such as olive oil, palm oil, coconut oil or soap butters such as mango butter, shea butter, etc.). These substances are mixed together until the "batter" thicks up and the batter is placed into a mold where it is covered with a towel and the soap goes through a gel phase (it heats up) and then cools down after about 1-2 days and the soap is unmolded, cut and placed on a drying rack and turned frequently to cure. After about 4-6 weeks of curing, it is ready to be used.
Info provided by my experience and from soap makers such as Edens Garden and The Nova Studio.