Personal identity is the concept that two configurations of particles (or computations, etc), each of which is a person, can be in some sense the same person. For example, you-right-now and you-yesterday are usually considered two instances of the person “you”, instead of two different people, even though they are distinct configurations of particles.
Philosophers have proposed many theories of personal identity, relying on various attributes like the two configurations being made from the same atoms, there being a particular causal relationship between the two configurations, there being a single extra-physical soul appearing in both configurations, the two configurations being sufficiently similar, personal identity not actually existing, and pretty much anything else you can think of.
The problem used to appear fairly straightforward, since no one had even considered the possibility that you could do things like create a copy of a person and run them on a computer. There were no boundary cases to suggest that our naïve intuitions about personal identity might be misguided. However, now that technological and scientific advances have suggested boundary cases to consider, these boundary cases give us opportunities for different theories of personal identity to disagree....