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 .
A mother's body, tonsils, appendix, arms, and legs all share the same genetic code. An unborn child within a mother has a genetic code that is completely distinct from the mother. The unborn child may also have a different blood type, or a different gender, than the mother.
The baby has forty-six chromosomes, twenty-three from the mother and twenty-three from the father. The combination of these chromosomes is completely unique (except in the case of identical twins). When there is a baby inside the mother, there are two distinct individuals, two distinct people.
John Jefferson Davis says this:
It is a well-established fact that a genetically distinct human being is brought into existence at conception. Once fertilization takes place, the zygote is its own entity, genetically distinct from both mother and father. The newly conceived individual possesses all the necessary information for a self-direct development and will proceed to grow in the usual human fashion, given time and nourishment. It is simply untrue that the unborn child is merely "part of the mother's body." In addition to being genetically distinct from the time of conception, the unborn possesses separate circulatory, nervous, and endocrine systems. 
Here are some other considerations that demonstrate the baby is a completely distinct individual from the mother:
It is obvious that the mother and unborn child are two distinct, unique individuals, and that the unborn child is not merely a body part of the mother.