I'll start with generators, seeing as they're the simplest case. As @zvolkov mentioned, they're functions/objects that can be repeatedly called without returning, but when called will return (yield) a value and then suspend their execution. When they're called again, they will start up from where they last suspended execution and do their thing again.
A generator is essentially a cut down (asymmetric) coroutine. The difference between a coroutine and generator is that a coroutine can accept arguments after it's been initially called, whereas a generator can't.
It's a bit difficult to come up with a trivial example of where you'd use coroutines, but here's my best try. Take this (made up) Python code as an example.
def my_coroutine_body(*args):
while True:
# Do some funky stuff
*args = yield value_im_returning
# Do some more funky stuff
my_coro = make_coroutine(my_coroutine_body)
x = 0
while True:
# The coroutine does some funky stuff to x, and returns a new value.
x = my_coro(x)
print x
An example of where coroutines are used is lexers and parsers. Without coroutines in the language or emulated somehow, lexing and parsing code needs to be mixed together even though they're really two separate concerns. But using a coroutine, you can separate out the lexing and parsing code.
(I'm going to brush over the difference between symmetric and asymmetric coroutines. Suffice it to say that they're equivalent, you can convert from one to the other, and asymmetric coroutines--which are the most like generators--are the easier to understand. I was outlining how one might implement asymmetric coroutines in Python.)
Continuations are actually quite simple beasts. All they are, are functions representing another point in the program which, if you call it, will cause execution to automatically switch to the point that function represents. You use very restricted versions of them every day without even realising it. Exceptions, for instance, can be thought of as a kind of inside-out continuation. I'll give you a Python based pseudocode example of a continuation.
Say Python had a function called callcc(), and this function took two arguments, the first being a function, and the second being a list of arguments to call it with. The only restriction on that function would be that the last argument it takes will be a function (which will be our current continuation).
def foo(x, y, cc):
cc(max(x, y))
biggest = callcc(foo, [23, 42])
print biggest
What would happen is that callcc() would in turn call foo() with the current continuation (cc), that is, a reference to the point in the program at which callcc() was called. When foo() calls the current continuation, it's essentially the same as telling callcc() to return with the value you're calling the current continuation with, and when it does that, it rolls back the stack to where the current continuation was created, i.e., when you called callcc().
The result of all of this would be that our hypothetical Python variant would print '42'.
I hope that helps, and I'm sure my explanation can be improved on quite a bit!
