[gjm11]

It's not expanding into anything (so far as we know), just expanding. A curved spacetime doesn't need to be, and so far as we know isn't, embedded inside a higher-dimensional not-curved thing in the way that a balloon is embedded in (more or less) flat three-dimensional space. (That sort of embedded picture is useful for the intuition, though.)

So, you might be asking, whatever does it mean to say that space, or spacetime, is curved? It means, e.g., that if you measure the radius and circumference of a small circle very very very accurately, the relationship isn't C = 2 pi r. If the circumference is "too small" then space is positively curved there, like a balloon. If it's "too large" then space is negatively curved there, like a saddle. The curvature might actually be different depending on the orientation of the circle, but if space is the kind of thing we think it is then you basically only need six numbers to tell the whole story at each point of space. (In two dimensions -- the surface of that balloon -- there's only one possible orientation for the circle, so you only need one number to describe the curvature at each point.) Note that you can measure this thing without needing to go "outside" space: the radius and circumference are measured within the space. Distances along the surface of the balloon.

Yup, the universe could be curved and topologically nontrivial in all kinds of interesting ways: it doesn't have to be like the surface of a balloon. (There's some intriguing but inconclusive evidence suggesting that the large-scale geometry of the universe is a "Poincare dodecahedral space": take a dodecahedron, and apply some magic to its faces so that when you try to leave it by one face you come back in by the opposite face.) And yes, light travels along the shortest paths it can within the universe, and doesn't take out-of-the-universe short cuts even if there are any.