Thanks.
If you just want to get a conceptual insight, try the excellent Wikipedia article on manifolds, which is fairly non-technical and introduces the concept of what a manifold is and why you would want to use one as a mathematical tool.
http://en.wikipedia.org/wiki/Manifold
In its barest essence a manifold is a way of reductively mapping a complex geometry. Knowing how to construct one, you can then deconstruct a complex multivariate problem (eg the mathematics of 10-dimensional strings) into simpler geometric problems, using the mathematics of manifolds to escape the hideous logical problems resulting from boundary conditions.
Consider an unbounded plane, ie normal cartesian space. You can roam around forever in the X-Y dimensions, boring. Now, suppose it's bounded; for example, mapped onto a finite sphere. This is one case of a 2-dimensional manifold (assuming a perfectly smooth sphere). It turns out that geometry still works perfectly well even though it's not flat. Georg Riemann was out to prove Euclid's parallel postulate by doing a reductio ad absurdum, and showing that if parallel lines could meet then all of geometry would cease to make sense. To his surprise, it didn't, and he found himself the unexpected inventor of spherical geometry.
Now consider a different kind of two-dimensional manifold: the world of Pac Man. Pacman lives in a rectangular world, but he can exit one side of the rectangle and re-enter at the opposite. On some maps he can also exit at the top and re-enter at the bottom.
But this isn't mappable to a sphere. Rather, imagine a cross shape, like a +. Please think of the arms of the cross as having thickness, rather than just 2 lines; better yet, draw a cross consisting of 5 squares on a sheet of paper and cut it out. Draw a line around the middle square and put a little pacman in it. Now, take the top and bottom squares and tape their outer opposite edges together. Do the same with the left and right squares. You get something resembling two cylinders at right angles to each other. Pacman lives on the little square in the middle where he eats pills and is chased by ghosts. He can leave one side of the square and travel via the curved path on the paper to the other side; he does so by literally traveling through another dimension - that is, he teleports (oooh!). It's important to remember that although a direct path along the curved loop of the paper is 2x the distance across pacman's rectangular world, as far as pacman is concerned it's actually zero. When he exits on one side, he doesn't disappear from the game screen for 2x the time it normally takes him to traverse the maze in a straight path, he reappears instantly on the other side. You could represent this in a simpler fashion by using a torus (donut shape), which is yet another type of 2-dimensional manifold, one on which pacman slides around on the surface.
In fact, this would be a much better way to represent it (because it gets us out of the annoying zero-length weirdness), but I offer the idea of the linked cylinder because it's worth considering that if you turn it upside down, there is an equivalently-sized opposite space. That's equally true of the donut (the inside has the same configuration as the outside) but less obvious. Pacman can never enter this opposite space unless he finds a way to go around the edge of the paper, which is another dimensional shift again, over and above his opposite-edge wanderings. Is there an anti-pacman there, called namcap? Maybe - or alternatively, you could imagine that when pacman eats a power pill, the ghosts become oppositely charged and are suddenly vulnerable to him - something we might not have thought of if we were looking at the game taking place on the outer skin of a torus.
So...manifolds are a set of handy mathematical tools for messing around with contiguous multi-dimensional spaces. String theory somewhat resembles a 10-dimensional game of pacman where both pacman and the ghosts are travelling at varying speed and can't eat each other except when their speeds are matched. When you see pictures of C-Y manifolds they're only slices of the manifold from one perspective. We can see pacman's whole world at one because he lives in 2 dimensions and we have the advantage of living in 3. To pacman we would seem like beings of infinitely variable shape and number, as our bodies could be sliced from an infinite variety of angles. So if your hand intersected his 2-d world, it would appear as a circle, followed by 3 circles, followed by 5, which would then merge together into a sort of oval (going from your fingers towards your wrist). Similarly, the cool C-Y shapes on the math pages are 3-dimensional slices of a 10-dimensional conceptual space, the overall shape of which we cannot meaningfully speak about except through mathematics, and which are impossible for us to really appreciate.
This is, of course, 1% mathematics and 99% bullshit. Not responsible for lost property left in other dimensions.
Edit: oops, I just realized it reads as if I were still talking about the torus, in which case inside out would be correct (but there would be no edges).