I only know of two types MIL-O-7808 and -23699. Is there anything else ?
Here is a general answer for a general question.
Function of oil is three-fold for both piston and turbine engines:
- Lubrication - reduction of friction;
- Cooling - carrying away of heat; and
- Cleaning - carrying away of contamination.
Oils used in jet engines must be able to withstand temperatures that are much hotter than piston engines.
Here is a link to some of the lubricating oils used in navy jet aircraft and the various specifications.
Piston aircraft engines tend to be air-cooled and are therefore manufactured with larger tolerances. This results in an engine that uses up oil (burns it). Most POH of piston aircraft that I've seen call for ashless dispersant oil, which refers to an oil that has non-metallic additives and has the ability to suspend combustion particles.
Here is an article that describes piston aviation oils.