Human life is an unbroken chain leading back through time to the point where the first human was conceived. That, however, is not the real point of the question asked in the OP. The question really had to do with wondering when a new, individual person begins to exist as a way to get into the abortion debate.
The scientific and intuitional position would seem to be that a person begins to exist at the point that something unique to people exists. Cells, organs, tissues are all part of what makes an animal, but again, we're talking about a "person."
What makes people different from animals is their mind. The self-awareness, the ability to learn, to speak, to reason. This is the thing that, once lost, is when we decide that a person is no longer alive, despite the body perhaps being kept alive artificially. It only makes sense that a person begins when this ability begins. It has a certain logical symmetry to it.
Now, since we cannot pinpoint the exact moment a fetus or baby becomes aware, gains the ability to reason, we can decide to err on the side of caution. We know that these abilities can not exist without the brain, so the creation of a functioning brain is a prerequisite for any of the abilities that are used as a determinant for personhood. Before a brain: no person. After a brain: perhaps a person.
Now, some people, mainly religious ones, will talk about a soul. Firstly, there is no evidence for such a thing, but even if we assume one exists, the attributes given to it are the same as the ones I listed as necessary for personhood, ability to reason (especially morally), self-awareness, etc. If this soul is an immaterial aspect of the human person, we have to ask, how does it interact with the physical such that it has any bearing on when a person has been created and should not be killed. If it doesn't interact with the physical, then it has no bearing on when a person can or cannot be killed as it would seem to make little to no difference to the soul.
If the soul interacts with the physical, and has the attributes listed above, the only logical conclusion is that it, too, requires a brain. Alterations to the brain are the only things that affect the attributes given to a soul, and so the brian must be the conduit through which the soul works. Again, no brain: no soul: no person.
And all of this combined, has no bearing on whether abortion can be performed after a brain is present because even then, we are talking about two persons, both with rights. The mother's rights do not magically disappear just because a new person gains them.
So, in summary of my summary, no matter how you slice it, a brain seems to be necessary to personhood. After a brain is formed, you have a person, but that still doesn't necessitate the position that abortion is immoral in all cases.