The oils are triglycerides. A triglyceride is made up of a glyceride ion attached to three fatty acid chains. When the caustic is mixed with the triglycerides, the fatty acids break loose from the glyceride ion and join with the sodium part of the sodium hydroxide. That forms soap. The glyceride ion joins with the hydroxide part of the sodium hydroxide, turning it into glycerine, which goes free.

So here is a short version of soap making:

[ Oils + Caustic ] yields [ Soap + Glycerine ]

The rest of the details, such as how much oil can react with how much caustic, and how you process the soap as the reaction proceeds, will all be left to those who want to do a little research. There are many good books on the subject.