(Update: just to clarify, throughout this post I use 'person' not in the philosophical sense, but merely as synonymous to 'human')
It strikes me as obvious that a single-celled zygote is not a person. It seems similarly clear that there is no relevant difference between an 8-month old foetus and a newborn baby. So when does the magical transition to 'personhood' occur?
It's almost like a (sort of) sorites argument - e.g. the paradox of the heap:
1) A single grain is not a heap.
2) If n grains is not a heap, then neither is n+1 grains.
3) Therefore, [insert absurdly large number here] is not a heap.
Of course, this is not a particularly strong analogy, because all sorts of important developments take place during foetal development (eg first heartbeat, brain activity, pain sensitivity), any of which may be used to draw a line between "clump of cells" and "person". But any such partition seems just as arbitrary as those previously discarded. What are we to do?
"Is a foetus a person?" - It's a yes-or-no question, right? Maybe not. Maybe the best answer is "sort of".
Perhaps there is no defining characteristic of 'personhood', no perfectly necessary or sufficient conditions for analysis to reveal. Instead, it could be a set of loosely-connected concepts, what Wittgenstein called "family resemblances".
Take the concept "game". There isn't anything all games have in common. Many are competitive, but not ring-around-the-rosie. Most are played for fun, but not professional sports. Often they involve many players, but not solitaire. So we have this fuzzy idea of a 'family resemblance' - each instance of a 'game' tends to have something or other in common with the other family members, but they may have many differences also.
So I think that 'personhood' may be another category which involves family resemblances. Most people have 46 chromosomes, but genetic disorders (e.g. Down's syndrome) may alter that without making you a non-person. Rationality? Empathy? Ten fingers and ten toes? We could come up with counterexamples to any of these suggestions. There simply is no single defining characteristic of humanity.
What are the implications of all this? Well, I think it shows that there is no metaphysical 'essence' behind the concept of 'person'. It is not a natural category (perhaps there are no natural categories, in this sense?). It's an artificial concept we made up to help us make sense of the world. To decide whether a specific object should be called a 'person', we simply judge whether it has enough of the relevant family characteristics, for it to be useful for us to class it as such in this particular context.
In other words, 'personhood' is one of those interesting concepts that must be decided, not discovered. We could know all the empirical facts about foetuses (their biology, neurology, psychology, etc), and still be at a loss as to whether they are persons or not.
In terms of the abortion debate, what we're asking is whether foetuses should be given the legal rights (particularly the right to life) normally accorded to persons. Within this context, to ask whether a foetus is a person is simply to ask whether a foetus should be given those rights. That is, the status of a foetus (as human or not) is not a part of the abortion debate, but a consequence of it.