http://en.wikipedia.org/wiki/Ape#Historical_and_modern_terminologyHumans are apes. No doubt about it.
Evolution is a process through which certain genes become more prevalent in a population. If the population is separated from other populations of the same species in some way, the differences between genetic content of the populations builds up over time until it becomes impossible to have fertile offspring with members of the different population. At that point, the separation between the populations is permanent and there is no natural way for them to recombine into one, and the populations are declared different species. These species can split again, and again, and again, but never grow back into one.
This means that, if you map out these populations in time and genetic distance, the result looks like a tree: branches break off, but never come back together.
This means that the most convenient way to classify groups is by the last common ancestor (population). Having that last common ancestor as your ancestor is a *sufficient* criterion for making a clear consistent group in any animal tree of life.
Humans have a common ancestor. Apes have a common ancestor. The common ancestor of humans has the common ancestor of apes as an ancestor. Therefore humans are a subgroup of apes.
The human common ancestor also has the common ancestor of primates, mammals, tetrapods, vertebrates, chordates, and animals as an ancestor, and therefore humans belong to each of those groups.