I wonder if "ape" is the generic, more general term than "monkey". Can one say that all primates (including monkeys) except lemurs, humans and some other few species are apes?
There's technical usage and there's common usage.
The technical usage is determined by the biological community and there the usage states that chimps, gorillas, humans, and gibbons are apes, and that there is a distinct category of monkeys that split into New World (spider monkeys, etc) with tails) and Old world monkeys (e.g. macaques and baboons). That is, monkeys and apes are separate groups, a monkey is not an ape and an ape is not a monkey.
As to common (informal, non-technical usage), monkey includes most primates: chimps and baboons and the canonical tree swinging monkey, possibly gorillas but not humans at all. At the zoo, in the great ape house, most people will point at the animals and say 'look at the big monkey' and some kid will say 'that's not a monkey, it's an ape'.
So if anything, informally, it's the other way around, 'ape' is used as a kind of monkey. But don't use that around anybody who actually knows anything.
But technically apes and monkeys do not overlap (though obviously related as primates).
Ape refers to the clade of animals belonging to the superfamily Hominoidea, which is the same superfamily that contains Homo sapiens. Humans, in other words, are biologically apes.
Monkey, on the other hand, is not a clade, since there are two different groups, the Catarrhine or "Old World" monkeys (superfamily Cercopithecoidea, found in Africa and Eurasia only) and the Platyrrhine or "New World" monkeys (found in the Americas only -- these are the ones with prehensile tails). The apes are related more closely to the Old World monkeys.
Consequently, if anything, ape is a subset of monkey, rather than the other way around. Though that's not strictly true, either. Check the tree diagram on the "Primates" Wikipedia page that tchrist mentioned to see the details.
Simians is the term I would use for the "higher" primates. Simian means monkey-like, and the term does correspond to a scientific classification, Simiiformes. Primates is arguable more common, but if we are to be as scientific as the previous answers demand, we must note that primates include lemurs, tarsiers, and other animals which many people would not understand to be closely related to monkeys and apes, as wellas humans.
In everyday speaking, most people will use apes and monkeys interchangeably as the umbrella term, either out of ignorance of the scientific distinction or for the sake of simplicity. This is loosely comparable to calling all dogs hounds or referring to horses as ponies— it may be incorrect, but in non-academic arenas, it gets the sense across.
This is really a taxonomy question, but since there is no Taxonomy.SE...
Both belong to an order commonly called Primates. While this isn't technically correct, you can consider primates as being split into "simians" and "prosimians" (these words aren't used a lot, but sometimes you will hear them in conversational English). Mostly "prosimians" are lemurs.
"Simians" are generally split up into monkeys and apes. They are separate things. This is important because if you call an animal that is considered an ape a "monkey" (or visa versa), some brainiac is bound to correct you. That's annoying enough IMHO to learn the difference. :-)
Monkeys are a rather diverse grouping. Mostly they are smallish tree-dwelling primates (who are not lemurs), although Baboons are monkeys, are rather large, and live on the African savannah.
So generally what you do is learn what Apes are, and you are fairly safe in calling all other primates (not found on Madagascar) "monkeys".
So what are Apes? They are Gibbons, Siamangs (collectively known as lesser apes), and Orangutans, Gorillas, Chimpanzees, Bonobos, and Man (collectively known as greater apes).
Humans of course are a special case. There are those who (generally in my experience for religous reasons) insist Humans are not animals, and thus certianly aren't apes. They tend to feel very strongly about this, so I would advise against referring to humans as "apes" either, unless you know for a fact your audience doesn't include any such people.