I don’t understand what you’re asking about here. Are you thinking a core is something you get rather than something you give? (See below.)
In a new one, every single piece is new. In a rebuilt one, normal wear items are replaced (e.g. brushes and bearings), some parts may be reconditioned (e.g. commutator slip rings), parts that seldom if ever fail are re-used (e.g. case, stator, rotor). New alternators are certainly – and rightly – perceived as better than poor quality rebuilds, but I question if there’s much practical difference between new and high quality rebuilds. I have a very low failure rate with high quality rebuilds.
A core is your old part, turned in for credit when buying a rebuilt part. The core is then sent to the rebuilding plant to eventually become a rebuilt unit for someone else to buy later. [ETA: A core is like an exchange, turn in your old one plus some money and get a refurbished one.] New parts are sold without expecting a core from you*, as the manufacturer of the new part is not rebuilding anything. As far as parts stores, I haven’t dealt with NAPA for years (not one close by), but always thought they had decent quality. I’ve been dealing with Carquest and O’Reilly and quite satisfied with their quality. I would be leary of places like Pep Boys and Autozone, as my perception is that they target the do-it-yourselfer who is more concerned with low price than with high quality. Note, however, that NAPA, Carquest, and O’Reilly want to compete with these places and so offer lesser-quality lines in addition to their high-quality lines. Stick with their top-grade stuff and you should be fine. Buy anyone’s budget-grade stuff and there’s a noticeably higher risk of getting a faulty part.
*I have heard of some new items requiring a core to be turned in, but these are specialized cases.