Saponification is the reaction of a metallic alkali (base) with a fatty ester to form soap. Saponification is the making of soap.
Lye is a form of sodium hydroxide which is a caustic base.
Animal fat is a fatty ester in the form of triacylglycerol. The alkali breaks the ester bond and releases the fatty acid and glycerol.
The soap is salted out by precipitating it with sodium chloride.
Saponification is often generalized to refer to the breaking of ester bonds using a caustic alkali.