What makes a person the same over time? Parfit used Locke as a starting point: It's a matter of continuity of memory: I am the same person as my younger self because there are overlapping chains of memory. Even if I and earlier me don't share any specific memories (Locke's actual criterion), there's a chain by which old me had a set of memories, then a day later lost some and gained some, and so on going forward to the present.
This reasonable sounding view has interesting consequences when you consider sci-fi examples: If someone makes a copy of your memories somehow, whether to teleport or to clone or to upload you to a computer simulation, then that copy would literally be you: You live on through the copy even if your original body is destroyed. However, identity is by definition a one-to-one relation: What if more than one copy is made? Are those copies and the original all 'the same person" even if they're now running around living independent lives and gaining new memories? They're all continuous, and so numerically identical, with the pre-copied self according to the theory, so by transitivity, they should all be identical to each other, but they're clearly not.
The main alternative theory is continuity of physical substance, specifically the brain, and Parfit thinks this option ("the physical criterion") is even worse. According to that theory, in the case of teleportation (where the original body is destroyed as a copy is put together at a remote location), you simply die. Yes, someone qualitatively identical to you ends up at the other end of the process, and it would think it was you and continue to pursue your projects and be a part of your human relationships, but it wouldn't really be you in the sense of numerical identity; the original brain was destroyed! Parfit's argument against this view uses a different sci-fi example: Imagine a brain surgeon is replacing small parts of your brain one by one with identical ones. The physicalist here is committed to the view that at the beginning and end of the process, it's a totally different person, because the entire brain has been replaced. It's also key to the notion of personal identity that two persons are either the same or they're not. There's no "sort of." Just as a continuous chain of memory constituted the view descended from Locke ("the psychological criteria"), a continuous chain of brain matter constitutes on the physicalist view the same person over time. Even if past-me and present-me don't have any of the same neurons, the replacement was gradual, so it's still me.
But this means there's a contradiction in the brain-surgery case. They physicalist has said no cloning, right? But the brain surgery is a gradual conversion of someone into their own clone, so the overlapping-chain character of this transformation should mean that either in this case, the clone really is still you, or at some point there was a change in personhood. But at what point? The concept of personal identity says that it's all or nothing: It can't be the case that cases in the middle are "sort of" the same. There must have been some middle point in which replacing a single neuron suddenly made you into a whole new person, but this seems implausible, for one because we could never figure out which exact point that was.
The third main view Parfit considered was the traditional religious one, best known to philosophers through Locke, that is really the source of this all-or-nothing logic, which is continuity of soul or spiritual substance. As Locke already argued, given that we can't actually sense such a soul, this seems problematic, because then how do we know that, for instance, our soul isn't being replaced every day? It's the memories, according to Locke, that are the sensible indicator of continuity of mind. If we consider the cloning and teleporting examples, then either the soul itself would have to be copied or split (what would this even mean?), or more likely, the soul sticks to one body and the copies would then be soulless, or have new souls? According to Parfit, we can't entirely rule out this view that personal identity is continuity of soul, but it doesn't seem to make a lot of sense. If we had real evidence for it, like solid evidence of reincarnation whereby people can remember their past lives, then that would be good evidence that there really are souls that can attach to different bodies, and that they can carry psychological characteristics and constitute the real basis of personal identity, but we have no such credible evidence. Imagine how much easier it would be for archaeologists if past-life rememberers could direct them to the appropriate digging sites! But that has never happened.
Parfit's conclusion from considering all these views is that none of them work, that the notion of personal identity is incoherent. However, this doesn't lessen the importance of the question, for the person who's contemplating being teleported, "will I die?" According to Parfit, strictly speaking, yes, you will. However, the continued existence of your identical clone with all of your memories is just as good as living on. Moreover, even during ordinary, day-to-day life, there's less continuity than we might think, and so more precisely, it's not that living on through your clone is just as good as ordinary persistence, but that ordinary persistence is just as bad as being cloned. This conclusion is supposed to have Buddhist ethical implications. If personal identity is a concept that ultimately breaks down and is not really the thing that we're concerned with in our existential wonderings, then extreme selfishness doesn't make sense. You are not so intimately connected with future-you and past-you that these connections vastly outweigh your connections to other people. Your concern for future-you is not fundamentally different than your concern for the welfare of others. If clone-me continues my projects and relationships, that's pretty much all I want out of life anyway.
Image by Solomon Grundy. Audio editing by Tyler Hislop.