The problem of personal identity is the question about how we define a person. In other words, what makes me me? How do we decide if there is some person who is the same person as me? Locke answers this problem by saying that personal identity consists of autobiographical memories. Reid shows that this theory will violate the transitivity of identity. Parfit's contribution to the literature on the problem of personal identity is to note that strict numerical identity might not be the appropriate concept. Rather, he talks about survival. The question of whether you survive is the same question as whether there is some person who is you. In the links below, I discuss the difference between identity and survival in more detail.
Parfit on PID.
Identity vs. Survival.
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