Today I gave my lecture on mammal diversity and evolution in the 4th year vertebrate course. We have been talking a fair bit about paraphyletic groups, common vs. scientific names, and so on. Within this context, we explored the issue of whether we're "descended from monkeys", by taking a look at a phylogeny of relevant primates:

The issues that we noted were:

1. "Apes", as defined as orangutans, gorillas, and chimps, but not humans, is paraphyletic. In other words, either "apes" is not a scientifically defensible term or else it must include humans.
2. "Monkeys" is paraphyletic, and in particular Old World monkeys are more closely related to "apes" than they are to New World monkeys. (Also, humans and Old World monkeys are equally closely related to New World monkeys).
3. We are not descended from any modern "monkeys" or "apes", rather we share common ancestors with them. (In that sense, the answer is NO to whether we're descended from monkeys).
4. The last ancestor shared by all apes (including humans) would itself probably have qualified as an ape. (In that sense, the answer is YES we are descended from an ape, but not any of the modern species).
5. For "monkeys" not to be problematic, it would have to include apes. In that sense, we would be apes AND monkeys. (And, for that matter, we're also lobe-finned fishes). As above, it may very well be that the ancestor of all monkeys and apes (the very bottom node on the phylogeny) would have been considered a monkey, and therefore YES we are descended from a monkey (but again, not any modern species).

After class, one of my students emailed me a link to this video, which explores the issues nicely. The author takes a cladistic approach and concludes that we are descended from monkeys for the reasons listed above.