Paul, in scientific terms apes are primates exclusively from the Old World (Eurasia and Africa) and distinguished from other primates principally by a wider degree of freedom of motion at the shoulder joint. Scientifically this group includes humans. The so-called great apes, or the
Hominidae if we're being more scientific, are a family of primates that includes eight extant species in four genera:
Pongo (the 3 orangutan species);
Gorilla (2 gorilla species);
Pan (2 species of chimpanzee) and Homo, which includes modern humans and their extinct relatives (for example the Neanderthals) and their/our ancestors, such as
Homo erectus etc.
Apes are a sister family of Old World monkeys, the
Cercopithecidae. Old World monkeys are unlike apes in that most have tails (the family name means "tailed ape") but these tails are never prehensile.
New World monkeys (there are five families:
Callitrichidae, Cebidae, Aotidae, Pitheciidae, and Atelidae) all seem to be descended from an ancester of the current Old World monkeys with this line splitting off about 40 million years ago, since when they have themselves evolved and diversified, exclusively in South and Central America, to give the five extant New World monkey families. These New World monkeys typically have flatter faces than their Old World relatives and also typically (with the exception of howler monkeys) lack the trichromatic vision of Old World monkeys (and of Old World apes, including of course humans). New World monkeys are the only ones to have evolved prehensile tails.
That said, the strict physical distinction between apes, and Old- or New World monkeys is primarily based on dentition. Moreover the definitive relatedness of extant species is now usually determined through DNA analysis, but this of course is rarely an option when dealing with exinct, ancestral animals.
So scientifically speaking humans are apes, and apes are most closely related to Old World monkeys, and at a slightly lesser remove from New World monkeys.
Here's quite a good up-to-date chart showing the taxonomic relationships between various extant apes and monkeys based on DNA studies (the red lines were just to illustrate a specific point in the article from which I filched it and although it appears to place humans outside of the other apes, nevertheless the relationship is clear).