Let me begin by summarizing some past philosophy as I've found it. Locke proposed the following criterion: if you remember thinking something in the past, then you are the same person as he or she who did the thinking. Modern writers add that the memory has to be caused by actual experiences, rather than (say) a hypnotist. (See Perry for an excellent review and anthology of writings on this subject.)
A key problem in personal identity is duplication. If it's possible to make a copy of a person so that there are now two where before there was one -- and uploading would surely permit this -- then which one is the "real" person? Various answers:
I think (3) deserves a little more explanation. It relies on "fuzzy logic," an extension of Boolean logic which allows truth values to be partial rather than entirely true or entirely false. This can be applied to personal identity as follows. Two people are "the same person" to the extent that they share a common history. Therefore, you and I are not at all the same person. I and the ten-year old me are not quite the same person. You and the you of ten minutes ago are (almost) entirely the same person. Now, if I duplicate Bob into Bob-A and Bob-B, at the instant of duplication, Bob, Bob-A, and Bob-B are all the same person. As time progresses, however, Bob-A and Bob-B become less the same person as each other, and less the same person as Bob (i.e., them before the duplication). This decrease in sameness happens at exactly the same rate that you and I become less the same as our younger selves. Bob-A and Bob-B will always be somewhat the same person, because they share a common history up to the point of duplication. But as they experience separate histories from that point on, they become increasingly different people. In short: personal identity is a fuzzy concept.
Practically speaking, there would still be all kinds of legal and social problems with duplication. But philosophically speaking, it doesn't seem too difficult if we can let go of Boolean logic.
[For a thought-provoking analysis with a slightly different approach by Albert-Jan Brouwer, click here. Also see the Personal Identity Forum for a two-sided discussion of several interesting cases.]