Ah, the definitionof 'synthetic' is tricky, I'll come back to that. However, from a practical standpoint 'synthetic' lubricants use high viscosity index (VI) base oils. Conventional base oil has a VI of around 100, high VI is anything more than 120. As I said in my other post, VI is a measure of the change in viscosity with temperature, a higher VI changes less with temperature i.e, thins less when hot, thickens less when cold.
When synthetics first started entering the mainstream (e.g. Mobil1) they used something call polyalphaolifin (PAO) and a small amout of various esters. PAO is an oligimerization (just combining) of decene (C10 hydrocarbon) into higher molecular weight molecules to achieve the desired viscosity. Esters cover a wide range of materials but can be used to adjust the properties of the PAO. However, as base oil technology has advanced, there are now many sources of high VI base oil manufactured using hydrocracking targeting 120+VI.
Shell has a unique position though, they built a big natural gas to liquids plant in Quatar that makes base oil from natural gas using the old Fisher Tropsch technology (invented by the Germans in the 1920's and used in WWII to make fuel from natural gas). This base oil is in the range of 140 VI, very similar to PAO in VI. It's really quite a good base oil (no I'm not a 'plant' from Shell), it doesn't need any adjusting fluid,is less expensive than PAO, and actually I think performs better. PAO is used in some applications mainly due to its superior ultra low temperature flow capability.
Now back to the he formal definition of synthetic: it is that the molecules comprising the base oil are artificially made (synthesized) rather than being a distillation product of crude oil. Many years ago when synthetics started, all base oils were distillation products of some type. However, today, almost all base oil, 'conventional' or 'synthetic' are actually synthesized via hydrocracking and isomerization (i.e. break down the molecules and build them up with the properties you want). The only difference is the VI targeted in manufacturing. These base oils have no sulfur (which was a residual of distillation) and are very 'pure', they appear almostwater white before additives are added (additives tend to be darker). So by the formal definition of syntetic, every quart of oil you buy for your car could be called synthetic. Chevron alludes to this by saying it contains 'Isosyn', they were the leader in this change to synthesized 'conventional' base oil. By the way, these base oils ARE better than the old distilled ones.
So, synthetic is really a marketing definition these days and by no means are all of them the same. But keep in mind 'synthetic' base oil doesn't impart any miraculous properties, claims of 'molecular bonding' and 'stronger films' are mainly clever marketing. The wear, oxidation, corrosion, and deposit control of modern engine oils today is carried by the additive package. A combination of anywhere from 5 to 15+ descrete chemical compounds added to the base oil to meet the performance requirments.
The last question is what are these oils based on instead of oil. Well as I said, Shell uses natural gas. Almost all of the other fluids are derived from crude oil by one route or another, even PAO.