Each oil requires a certain amount of lye in order to saponify and messing with the amount can mean the difference between a beautiful bar of soap and a glob on your counter. It helps create a hard, creamy bar of soap yet, used all by itself can dry your skin. It produces a creamy soap with small bubbles and is very mild. Plus, it's packed with all sorts of goodies such as vitamins and minerals. Popular soaps made with olive oil are Castile soaps which are the cream of the crop, in my opinion.
Sunflower oil is another popular oil which comes from sunflower seeds.
Its inexpensive and easily attainable which is a bonus. It is good for all skin types but tends to make a soft bar of soap which is slow to trace. I look forward to trying white palm oil as I understand it makes a very hard and long-lasting bar of soap. Shortening is another popular base oil. Good ole Crisco.