I don't want to speak for all pro-life people, but since this discussion frequently gets dominated by religious people against atheists I feel compelled to. I'm a gay, atheist, pro-life, man (judge me as you will).
It's not even that the "right(s) of the fetus under consideration is more significant than the right of the woman", rather that I'm judging the quality of one right to another. The right to life of one individual will always outweigh the right to convenience of another. The vast majority of abortions in the developed world are those of convenience. Those that aren't can be easily classified into those that involve biological necessity (i.e. woman's compromised health, deformity of birth, etc.), and those that involve psychological necessity ("I was raped!", "Dad is no longer in the picture!", "What will my family think!?", etc.) While I sympathize with those of the latter category, I cannot in good faith support their desire to murder their offspring.