I can tell you about both ends and a bit in-between. At the beginning, a fertilized egg is just a template. Genetic code that is basically a "how-to" to build a human. And seeing as that plan changes in the womb based on environmental factors, it's more of a rough draft of a plan. 1st-draft blueprints, let's call it.
The woman's body provides the resources needed to build the blueprints. It does it slowly, and does it oddly. We are a history of our evolution, and this shows in our embryonic development/blueprint building phase. It takes a pretty long time for the blueprint to look different than most other mammals, actually. Closer to birth you get, the further down the evolutionary chain you go. As our big brain is pretty much our huge unique feature when compared to our closest relatives, this also means it is the last to develop. This happens around the third trimester, or month 7. As we are our brain, it's pretty safe to say the person isn't a person until maybe probably around this point. This is also around the point when a child starts becoming viable (capable of living outside the womb) for similar reasons.
Usually it's near or before this point that most laws are "nah, no abortions past this point". which most everyone is fine with. As for what makes a "baby", I guess that depends on whether you consider a person is their brain or if they're something more...supernatural.
