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.
As well as suggesting boundary cases with which to differentiate different theories of personal identity, modern science also gives us some clues as to which theories are correct. For instance, evidence from neuroscience suggests that cognition is entirely physical, which contradicts theories of personal identity that rely on an extra-physical soul. Experiments from quantum mechanics show that particles don't actually have individual identities; that is, if there are two electrons at time 1 and two electrons at time 2, there does not exist any fact of the matter as to which electron at time 1 is the same as which electron at time 2. This rules out theories of personal identity based on being made of the same atoms.
Personal identity may at first sound like just an abstract philosophical issue with no practical consequences, but in fact, there are practical reasons to understand personal identity. For instance, common objections to cryonics and brain uploading hold that anyone who is woken up from cryonic suspension or whose brain is run on a computer would not be the same person they were before the operation, and that the operations thus fail to continue the person's life. Such objections are generally based on theories of personal identity that can be shown to be false or incoherent by modern science, as explained in the sequence on quantum mechanics and personal identity. It is already possible to sign up for cryonics, and whole brain emulation may be possible in the future, so it is actually possible to act on an understanding of personal identity. Once whole brain emulation is feasible, it would also be possible to easily copy and modify brain emulations, which would offer more challenging questions about personal identity.