Some claim that the unborn is just a part of the mother's body, like her tonsils or appendix. For example, philosopher Mortimer Adler claims the following:
[The unborn is] a part of the mother's body, in the same sense that an individual's arm or leg is a part of a living organism. An individual's decision to have an arm or leg amputated falls within the sphere of privacy —the freedom to do as one please in all matters that do not injure others or the public welfare." 
Note: This page heavily utilizes Randy Alcorn's terrific book, Pro-Life Answers to Pro-Choice Questions .
If the unborn is merely a part of the mother's body, then the death of the mother should result in the death of the unborn. However, this is not the case. There have been situations where the mother has died, yet a doctor was able to deliver the baby afterwards:
Furthermore, it is possible for a child to be born several months after the mother has been declared "brain dead." 
That the unborn can survive while the mother dies demonstrates that the unborn baby is a distinct person from the mother.