It's not that we are descended from monkeys, but that we share a common ancestor with them. Just like we share a common ancestor with the first forms of life and every other form of life. Read "The Greatest Show on Earth" by Dawkins to better understand this.
The way he explains it, life forms can be imagined on a 'evolutionary ladder', and life forms that we know of today all diverged from the path at a cetain point and developed on their own. I'm paraphrasing of course.
It's more a semantics argument than a biological one. The lowest clade in which we (man) and they (old world monkeys) share a common ancester is Catarrhines (which means simple-nosed). One step above that we all merge with the new world monkeys (Platyrrhines, flat-nosed) in the clade Simiformes (monkey-shaped). So, if a basal Simiform or a Catarrhine can be call a monkey, then we shared a monkey for an ancestor, but note that he will be neither an old or new world monkey.
In general, the English equivalent animal words are usually paraphyletic (not all descendants of such have share the name, such as "dinosaur" doesn't usually include birds). "Monkey" is paraphyletic, normally, as it doesn't include man. Biological systematics doesn't strictly allow for this; all clade names must be monophyletic (the name covers
all descendants). So, we're all simiformes, and all catarrhines. Does that make us all monkeys? Semantics.
Same goes for being apes, as well. If a basal Hominidae (commonly, a hominid, though the use is debatable) is an ape, then we are all apes. One step further up is the clade Hominoidea, which also include Gibbons; same goes for that clade. All us people are Homo, Hominini (or homonin or hominin), Homininae, Hominidae, Hominoidea (and that is not the same as humanoid), Catarrhinea, Simiformes, Haplorrhini (dry-nosed!), and primates, and about 30 other named clades all the way to Eutheria. Above that it is no longer a tree, but rather a free-for-all of life.