Two truths seem incontestable:
(1) The human embryo or fetus is a living organism of the species Homo sapiens, and elective abortion entails the killing of this unborn human being.
(2) It is wrong to kill an already-born (or at least developed past the infant stage, so that Peter Singer-types will agree) human being for the reasons people have elective abortions. For instance, a father may not kill his five-year-old daughter to relieve himself of the burden of caring for her.
The first is a plain factual matter. The second is a moral truth so basic that it is only rejected by psychopaths.
Given these incontestable truths, I think there are two ways elective abortion could be morally permissible. Sophisticated defenders of abortion tend to claim one or both of the following:
(1) There is a morally significant difference between the unborn human being and the already-born human being such that killing the former is permissible.
(2) The unique circumstances surrounding abortion (the unborn is inside of and connected to the pregnant woman for nine months) are relevant in such a way that the killing of abortion is morally justified.
A defender of the permissibility of abortion rationally must take one of these two approaches, it seems to me.
Most philosophers who defend abortion argue for option one. The problem is that there simply are not any morally significant differences between unborn and already-born human beings; size, level of development, environment and degree of dependency don't make a difference in terms of one's moral value and right to life. Indeed, all attempts to distinguish members of our species who merit full moral respect from those who do not, fail, for the proposed criteria are arbitrary, undermine human equality, and exclude obvious examples of valuable human persons. So the only remaining option -- given incontestable truth number two -- is that all human beings are fundamentally equal, and thus the unborn, like a five-year-old, deserves to be respected and protected. Option one fails.
A few philosophers argue for option two, offering some famous thought experiments. But it is not clear how any such approach can justify intentional killing, which is what abortion almost always entails. Moreover, even if the death of the unborn is foreseen but not intended (e.g., a hysterotomy), the special circumstances of pregnancy -- the child's vulnerability and dependency, the parent-child relationship and attendant moral obligations -- seem to make elective abortion impermissible. I have written about this second option -- a sophisticated argument from bodily autonomy -- here.
In short, both possibilities for defending abortion fail. Killing unborn human beings by abortion is wrong, just like killing older human beings for the same reasons.