Certainly, a living cell is alive from conception. However, bacteria are also alive, and no one on Earth has any problems killing them. Your question is therefore loaded, and ought to be "When does a foetus acquire full human rights?".
Since it is equivalent to a bacterium at conception, and a full human being at birth, the line should be drawn somewhere between the two places. I think it should change based on the purpose.
Abortions for clear-cut medical need (i.e. high chance of death or extreme disability in either the mother or baby) should be done as soon as possible after the result emerges, anything up until birth. Abortions for lesser medical needs that are not life-threatening (Down's syndrome, mother's psychological health, baby's welfare) should be done, again, as soon as possible after the result emerges, but would have to be more and more serious to justify later and later abortions.
Careful consultation between several doctors, the mother and in most cases the father should be done to determine, on an individual basis, whether a late abortion can be justified. Medical estimates of when pain-sensing or awareness starts should only be used to aid the process, not set hard lines or dates.
Unlike medical abortions, Purely social (I don't want one) abortions should not be allowed after a short amount of time after conception. That is a line the legislative authorities should decide, with the aid of scientific evidence. I would say about 10 weeks or so.
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For people who attempt to use religion to justify completely banning abortions, remember:
i) There is no scientific evidence for the truth of religious beliefs, and therefore:
ii) You can't force your belief on other people's actions.
So, you can continue to let religious ideas influence you, but you can't claim what anyone else is doing is wrong purely on religious grounds.
