three main types horizontal gravity separators; vertical gravity speapartors, cyclonic separators.
The gravity speapartors are most common; vertical ones take up less footprint space (useful offshore) but have a shorter residence time than horizontal separators...residence time is important in separating oil & water, so if you're separating oil and gas or gas and water, then consider a vertical separator.
Hydrocyclones use centrifugal force to separate reservoir fluids, it's a very elegant theory that I've used sucessfully countless times in mud cleaners on drill rigs, but I've never seen a hydrocyclone in action on a production platform.