Calabi-Yau manifolds (there’s not just one) are a type of mathematical concept, but they’re not “just” that as they do have applications in certain attempts to describe physics. The role they usually play in physics is to help us formalize the relationships that we postulate between the various internal variables that describe what particles are likely to show up at a point in space time. As such the theory often combines the six dimensions of a Calabi-Yau manifold with the four dimensions of the space-time that we are ‘inside’ to get a total of ten dimensions. But the extra dimensions are often either considered to be very small in some sense, or to have the part that contributes to the physics we see be just a slice through the whole thing. In the first case it makes more sense to say (as in James Bridgeman’s answer) that there’s a C-Y manifold at every point inside us (rather than vice versa), and in the second case that the entire space-time we live in is just a (4d) slice through the extended C-Y manifold (with other slices or “branes” perhaps corresponding to “alternate universes” of some kind). But neither of these cases is in any sense known to be true. So far it’s all just speculative construction of mathematical models that might eventually be shown to describe our actual physics.