The earlier stages of pregnancy are very different. Let's first of all dispose of the idea of “conception”, which is a religious idea that doesn't have any real biological meaning. The biology of human development begins with fertilization, which can't really be called an “event” because it is a complicated process in many stages that can be accomplished in different ways. But for the moment I'll concede the point and say that a zygote exists after fertilization. Once the DNA from the egg and sperm have combined, the newly-formed zygote then begins to divide into two cells, four, eight, and so on. At this point there aren't yet any specialized cells: they're all stem cells, and will only take on specialized roles as organs, nerves, and so on much later in the process of development (if they continue at all—many of them will just die off).
The process of development itself can take many turns. The majority of the time, in fact, the process results in nothing at all. Most fertilizations are simply flushed out with the mother's next menstruation and never develop. The woman never knows that any fertilization occurred at all. In those fewer cases where the zygote does make it through the tubes to implant in the uterus, its fate is still undetermined. It might develop into a person, or two people, or three, or half. Identical twins, for example, result when the multi-celled zygote splits at some point, and both portions go on to implant and develop into fully formed unique people (albeit with identical DNA). Identical triplets are quite rare, but also possible. Another even rarer possibility is that two different zygotes will merge at some point in their development and develop into a single fully-formed person with two sets of DNA. These are called tetragametic chimeras, and are often born with defects, but can also be born as perfectly normal infants who may not even know that they were the product of two different fertilizations.
This is where the “life begins at conception” argument falls down. Yes, a zygote after (and arguably before) fertilization is living, in the same sense that any of our skin cells or liver cells is living. They can divide and grow and contain a full set of genetic material. We can now grow skin and muscle tissue in a dish from a single cell. But the relevant question is not whether the thing is living or not, or even whether it is human. The question is “is it a person?”, in the sense of laws that make harming people an act of violence we detest, or is it merely a collection of cells like the skin cells we flush down the drain when we wash our hands, or blood cells that we donate to the Red Cross? What is it about people that makes them specially deserving of protection? A good way to answer that is to think of the twin case: why do we consider twins to be two people, not one? It's simple: each twin thinks and feels and dreams independently. Each has its own personality and its own desires and fears. It is untenable to argue that the single zygote that would later develop into these two people had any of those qualities. It had no brain, no nerves, no eyes, no ears. It had only the potential for developing those things—and we didn't know then from the zygote stage exactly what it might develop into. It might have become a person, or two, or half, or it might not.