This is an admittedly academic topic with few real world implications, but it is something I've devoted a lot of thought to anyway. Our place in nature is an intriguing scientific and moral question to consider.
The appropriate place to start is a discussion of taxonomy. Taxonomy is how we classify species. The premise of taxonomy is that we group species hierarchically based on how similar they are.
However, this raises the question of what we mean by similar. Do certain characteristics take precedence over others? Why?
The point is that grouping species together simply because they happen to have anatomical similarities carries a bias; it isn't objective, and it may not actually "tell" us anything about the relationship those species share. It is, in a single word, arbitrary.
For example, should a whale be classified as a fish since it swims or a mammal since it has mammary glands? Should bats be classified with the birds because they fly?
There is one sensible (i.e. objective) way to group species, however, and that is based on their evolutionary relationships. Evolutionary relationships reflect genetic similarity, which is an objective measure that actually tells us something meaningful about the species in question.
Applying this kind of taxonomy, we know to classify whales as mammals, and an analysis of their DNA confirms this. It isn't arbitrary at all.
Aside from the fact that whales being mammals is pretty dang cool, this isn't really a big deal. It isn't scientifically or morally controversial.
But that changes when we talk about humans.
Even before the theory of evolution was formulated, taxonomists knew to group humans with the primates. Within this order, it was also intuitive to say we had more in common with the great apes than the monkeys. But it is molecular data that revealed the real shocker . . .
Great Ape Phylogeny. Learn to interpret these
correctly for your own sake!
An analysis of the genes of these species has shown that humans and chimpanzees share more DNA in common than either does to a gorilla.
Intuitively, this makes very little sense--how can it be possible? In so many ways, we are not like them. Culturally, we developed agriculture, sophisticated technology, language, and inhabit the entire globe. Biologically, we've lost our hair, walk on two legs, and have a plethora of a other unique physical characteristics--even with respect to our closest relatives.
How such a small difference in DNA can account for such significant phenotypic differences is a question that boggles the minds of our best scientists. Yet the data itself is well established.
One implication of this is that it makes no sense to call Orangutans, Gorillas, and Chimps "apes" unless we also are willing to adopt that name for ourselves. In fact, we cannot legitimately apply ANY scientifically meaningful label to chimps and gorillas unless we apply it to ourselves as well because humans and chimps share a more inclusive relationship (note in the figure above that the lineage that led to Gorillas diverged before the lineage that led to either chimps or humans).
Some scholars (like Jared Diamond*) go even further. He notes that the genetic difference between humans and chimps is smaller than that between some species that are classified under the same genus (he cites lions and tigers as an example--so, unbelievable as it may seem, the genetic difference between a lion and tiger is greater than between a human and a chimp). Thus, he argues that chimps should really be called "Homo Troglodytes."

*
Jared Diamond's book. An excellent and highly
recommended read, although I have a problem with
the cover art since it invites the misconception that
humans evolved from chimps
Of course, there is disagreement with this perspective. My "Primate Behavior" professor, for example, sees no problem with dividing a humans and human ancestors from the rest of the great apes (referring to the former as "hominins and the latter as apes). He believes there are sound scientific and moral reasons for doing so.
I have tons of respect for this professor. He is a great teacher and well known in his field (he is the head of the Harlow Laboratory, the lab that pioneered the famous study that demonstrated a Macaque would choose nurture over nourishment, drastically altering our perspective on parenting.)
And on one level I agree with him. Humans are very clearly different than the other great apes. We are special. And our big brains and bipedal nature are fairly reasonable phenotypic markers of this difference.
Yet, if we believe the molecular data, the distinction remains entirely arbitrary. There is no way around it. I think it is justifiable (and useful) to use the term "ape" informally to refer to Chimps, Gorillas, and Orangutans, but we should never be fooled into thinking it "means' anything. Apes cannot represent a clade unless that clade includes us.
This situation should highlight the inherent bias of taxonomy (even among scientists). For my own part, I try to compromise. I do think we must consider ourselves "apes" (and if we find the connotations of that term offensive, then we should devise a new term). However, unlike Jared Diamond, I do NOT think we need or ought to include Chimps in the genus Homo. Evolutionary relationships are an exact science, but the definition of what constitutes a genus is not. Since we don't have an exact definition of "genus," it is reasonable to "fudge" it a bit to accommodate our anthropocentric view of the world.
What do you guys think about this? For the record, I don't expect many replies, but I hope at least some of you got something out of the above and give it some thought . . .