This would be very good to sticky. Thanks!
Group I is solvent extracted from crude oil. It's a mixed bag of hydrocarbons and in North America has fallen out of favor due to not being of very high quality compared to readily available base stocks that are improvements on it.
Group II is hydrocracked and dewaxed oil to produce more of the desired properties and it reduces unsaturated hydrocarbons, which reduces the tendency of the oil to oxidize and thicken out of grade. Most conventional oils in North America are made up of this.
Group II+ is an oil that would typically have very few unsaturated hydrocarbons and lower wax content. It may not be as fully processed as group III, but it is ahead of Group II for quality. Certain higher quality conventional oils use a higher proportion of this.
Group III is a heavily hydrocracked mineral oil. Most synthetic oils on the market are made up of this. The oil molecules are broken down through a variety of processes until they are much more like what has good lubricating properties, and removes or breaks down molecules with negative performance effects such as waxes. Pennzoil Platinum is the highest regarded group III oil on this site.
Group III+ is the extreme end of this, where the final product is basically nothing like any of the components that went into it, but it is still derived by heavily processing crude oil. An example of this would be Shell XHVI base oils. As well, oils are being developed on gas-to-liquids basestocks which would also produce extremely high quality group III bases, fundamentally synthesized but not falling into the definitions of group IV or V.
Group IV is PAO, which most consider along with group V to be a "true" synthetic. It is made up of base oils made from other hydrocarbons (often derived from natural gas) and is made specifically to be a lubricant, and does not have origin as crude oil. Mobil 1 and Amsoil are the best examples of these type of oils. They offer performance throughout extremes, resist oxidation, all kinds of fun stuff.
Group V is a blanket term for anything else, usually ester based oil. Red Line Oil is a common example of this.