In order to relate groups of organisms to one another, we must define what it is that separates one species from the next. In the case of humans, it is agreed that our closest living relatives are the chimpanzees. We share over 99% of our DNA with them! However, it is clear just by qualitative observations that we are very different indeed. These traits which differentiate one group of organisms from the next are termed apomorphies. An example of an apomorphic character is our "hairlessness", it separates us from the other primates. Those shared characteristics which group organisms related through a common ancestor are termed pliesiomorphies. An example of this would be hands with thumbs. It is these characters which align us with the rest of the primates, and the apomorhies that separate us from them.
Due to the space limitations on this site, I will concentrate only on those apomorphic characteristics that separate us from the other primates and are not explained well by the currently "accepted" theories.