Right to Lifers believe that the rights of an embryo/fetus always take precedent over the rights of the woman carrying it.
Pro Choicers believe that a woman has some rights that supersede those of the embryo/fetus for some portion of time between conception and birth.
The US Supreme Court in Roe v Wade (http://en.wikipedia.org/wiki/Roe_v._Wade) concurred with the Pro Choicers -- deciding that there is a period between conception and birth where a woman's rights supersede those of the embryo (and state).
In my view, the US Supreme Court decision was proper and fair. There is no good clean dividing line between non-existence and life and a reasonable choice by reasonable people is the best we can do. In this case the woman herself is in the best position to make that decision, as it is her body and her life.
We (mankind) have empowered ourselves to make decisions for ourselves rather than letting nature dictate -- by using knowledge and tools to change the world around us. Yes we chose to bite into the Apple of Knowledge and now we have the right and responsibility to choose our own destiny. For better or worse, this is the path that God (or whoever) has let us pursue.
We have enough people already in the world. It is not a requirement we have more and in fact we should even reduce our population. Mistakes are easy to make at conception. Nature itself causes "abortion" to happen if it determines that the embryo/fetus is not viable. For a portion of time, people should likewise have that option to choose for medical, economic, and life reasons. Not all women or families want or are capable of raising a child (or an additional one) -- or be willing to give it away for adoption. None of us have unlimited resources. We all must make hard choices at times. We cannot have unlimited life. In fact cancer is an example of unlimited life with no control.
The foundation of the US is based upon a balance between the rights of the many and the rights of the few. We cannot dictate one religious view on others.
Thus not only is pro choice correct for moral reasons, it is also correct for many practical reasons as well. There is not a simple black and white choice in this case. We have the power, we have the responsibility, we have the right for each of us to make our own choices and choose our own destinies.
The birth of each human has huge consequences for our world and deserves to be weighed carefully by those who are most responsible for that new life -- the parents and particularly the woman who is bearing the child. Roe v Wade is a reasonable compromise.