If we evolved from monkeys, why are there still monkeys?
First, let's reword that to make it a bit more accurate and meaningful: "if we evolved from non-human primates, why are there still non-human primates?" Better yet, let's further narrow the discussion down to why there are still chimpanzees, given that humans are directly descended from an animal that might well be classified as a chimpanzee if individuals were alive today.
The answer is speciation, a central concept in evolution. Speciation is the process whereby one species splits into two. For each species alive today, there has been at least one speciation event. The process whereby the human line split from that of our nearest living cousins was a bit messy, and might be more difficult for some people to accept because of our biased perspective: humans seem so different from (and superior to) chimps. So let's start out by briefly discussing a similar, but easier to understand speciation event: the split of polar bears from brown bears. This happened about 30 times more recently (and technically is still happening), and could be considered relatively "clean." Also, unlike humans, polar bears still live in the environment that they originally adapted to. We'll come back to humans a few paragraphs down.
The brown bear, which includes the grizzly bear and several other subspecies, has existed as a distinct species for about a million years. Approximately 200 thousand years ago, a population of brown bears split off to become polar bears. Most likely this happened when that population was isolated from the others by a glacier, during a period of climate change. Over a few tens of thousands of years, intense environmental pressures filtered out the bears that were the least adapted for life on the ice, resulting in dramatic changes in appearance and behaviour in a relatively short timeframe. At least one adaptation -- significant differences in molar teeth -- apparently came about just 10 to 20 thousand years ago.
What this means is that polar bears can be said to be "evolved from brown bears," while leaving brown bears still around, relatively unchanged from the ancestors of polar bears. Neither brown bear or polar bear is superior to or "more evolved" than the other, they are simply adapted to different environments.
While polar bears are classified as a separate species from brown bears, note that the speciation of polar bears is not 100% complete. Polar bears can interbreed with brown bears and produce a hybrid -- and they are known to do this, albeit rarely, in the wild. This means that in theory, the two lines could still merge back into a single species. However, if humans weren't around to destroy their habitat, it's a good bet than in a million years or so polar bears would be so far removed genetically from brown bears that interbreeding could no longer take place.
The speciation of polar bears is not so different from the situation some 5-7 million years ago whereby a population of "forest apes" were isolated in the the relatively treeless savanna. The new environment caused dramatic changes in the newly savanna-dwelling apes, eventually leading to humans. Forest dwelling apes are still around today, probably appearing and behaving quite similarly to how they did before the split, because they still occupy a similar environment, the forest. We call the modern-day forest apes "chimpanzees."
While we may consider humans to be "more highly evolved" than chimps, this is only because we place such a high degree of importance on things like intelligence. Chimps are much stronger and better tree climbers than humans, and probably have better immune systems -- making them more fit for life in the jungle than we are.
Of course, if you want to say that according to Darwin, humans should exist while "less evolved" animals such as chimps should not, consider that humans currently outnumber chimps by sixty-thousand to one, and that number will only grow.
The important point to remember is that just because one thing evolves from another, the original thing will not necessarily cease to exist. Obviously, if it did, we would only have one species on the planet today, and yet we have millions.
A note to the pro-evolution folks:
A common response I hear to the monkey question is "we didn't evolve from monkeys, we evolved from a common ancestor." I suggest that that argument is neither a necessary nor relevant, and I think this is supported by the polar bear analogy (as the common ancestor of modern day polar bears and their most closely related brown bear cousins was most certainly a brown bear itself). Whether our common ancestor with monkeys would be called a monkey (or even whether our common ancestor with chimpanzees might be called a chimpanzee) is not a particularly interesting question, in my opinion, and has more to do with human naming conventions than it does with actual science or the logic of evolution and speciation. Instead it only serves as a pedantic distraction -- a way of avoiding the question or insulting those asking it, rather than trying to understand and address what is confusing them.
If it is offensive to some to consider that we have monkeys, apes and/or chimps as ancestors, so be it. We also have other animals as ancestors (some kind of worm, maybe?) that would probably be seen as far less pleasant than a monkey.