What is the primary type of fat used to make your ordinary bar of soap? How about hand soap (the stuff in the pump)?
Most true soap is made from animal fat, primarily beef tallow and hog lard. Liquid soap may be a blend of soaps from solid fat and vegetable oils, possibly with a solvent (green surgical liquid soap has alcohol in it, IIRC), but most liquid soaps are actually synthetic detergent blends (“syndets”). Many of the fancy clear bar soaps are also syndets, sometimes with a little soap mixed in.
The difference between bar and liquid soaps is the type of lye used. Bar soap is made with Sodium Hydroxide and liquid soaps (bubble bath, shower gel, dish soap, etc) are made with Potassium Hydroxide.