First the generalities attempt at description:
PAO stands for Poly Alpha olefin.
Olefin is an older "common" name referring to a compound containd double bonds.
Alpha signfies that the double bond is to an alpha carbon.
Poly means that more than just one such bond is formed.
An ester is a an organic chemical that includes a functional group in which a carbon is double bonded to an oxygen, one other carbon and single bonded to yet another oxygen.
(these things are easier to draw than explain)
These terms refer to general classes of chemicals not all of which are good base stocks for oils.
BUT when used in lubrication chemistry describe classes of base stock that have many superior qualities.
Where PAO stocks are group IV and esters group V.
PAO base lube stocks are made by taking ethylene gas and conducting additions reactions in which, they add up to each other to form a base oil.
(a so called PAO base stock).
ester base stocks are often used too, for one they are great lubes and also have "natural" built in seal swelling properties, which can eliminate or greatly reduce the need to add swelling agents to an oil.
The "space" now left in the formulation can be used for other additives then, such as antifoaming, anti wear or more base stock.etc etc.
A group III base oil is a hydroisomerized stock of so called parrfin components of refinded oils.
It has excellent lubrication qualities as.
It os cheaper to make as a real syn as you just change a good base oil into a better one via isomerization.
A very good base oil, just not as good a full syns such as PAO/ester formulations.
Some "synthetic" oils use only or primarily group III base stocks (Castrol Syntec being the most notorious examnple).
thsoe are good oils, Syntec meets some of teh more rigorous industry specs.
Many maintain that PAO and/or ester based full synthetic oils are much better.
And I tend to agree.