They are generally talking about group III base stocks. Very briefly, Groups I, II and III are all 'dinosaur juice', Group IV and V are other things, though through a court case in the 80's Castrol won the right to call it's Group III dinosaur juice a 'synthetic'. Generally groups I, II and III all mix without issue, whereas true synthetic group IV and V don't. For the purpose of clarity, I refer to group III as mineral, although it can be legally marketed as synthetic.
How do you know what group you are using? There is no legislation to force oil manufacturers to specify, so when you're talking mixabilty, don't do it unless in an emergency, and you're going to change soon after. You can use MSDS to look for clues, but who does that? (no really I do but who else does?)
General rule of thumb- true synthetics, group IV and V synthetics for new engines. Group I, II and III Minerals for new or old engines. True synthetics tend to be 10w and thinner (not all 10w are synthetic), some might contain the words 'ester" (which can also be an additive in mineral oils- very expensive but good synthetic lubricant). In saying that, some 20w50 and 25w60 blends are group III, and can be legally marketed as synthetic, but aren't ;)
Why synthetics vs mineral? Marketing. Synthetics are better in some circumstance, minerals in others. But marketing will dictate that synthetics win everytime.
Which to use without a testing regime? Which feels best, gives better gearchanges, less vibrations, etc etc.