"Curled-up dimensions"? I'm a grad student in math, but I don't know as much about physics as I should.  I've read a handful of pop expositions of string theory, and they often refer to "curled up dimensions" and I've always wondered what this means.  Is this referring to any kind of standard mathematical object, e.g. a bundle of Calabi-Yaus?  (Actually learning some physics is one of my long-term self-improvement projects, but for now I'd be happy to demystify this word!)
 A: The basic idea doesn't have to be that complicated, and it isn't specific to string theory. We seem to live in a spacetime with four dimensions (three spacelike and one timelike). If there was a fifth dimension, say a spacelike one, you'd think we'd have noticed it. However, suppose that that extra dimension has the topology of a circle, i.e., spacetime is topologically a direct product of $\mathbb{R}^4$ with a circle. That circle has some circumference, which sets some length scale $\ell$. If $\ell$ was very small, we might not be able to detect the existence of the extra dimension.
If you like, you can add more than one extra dimension. (String theory requires 6 or 7.) Those extra dimensions can then have intrinsic curvature, which is another way of setting a distance scale $\ell$.
To a physicist, the next obvious question is what sets $\ell$. General relativity is completely invariant under any smooth change of coordinates, so the theory has absolutely no scale that is privileged over any other scale. For example, the spacetime at the surface of the earth has curvature that sets a certain length scale (which is how GR describes the gravitational field), but that scale isn't hard-coded into GR, it's due to the mass of the earth, which is an accident of the formation of the solar system.
The typical answer is that $\ell$ should probably be the Planck length, which is the scale at which you can't describe spacetime without using some theory of quantum gravity, which we don't yet possess. In most theories, the Planck length is $10^{-35}$ m. In some theories, known as theories with large extra dimensions, http://en.wikipedia.org/wiki/Large_extra_dimensions , it's much larger, and therefore possibly accessible by experiment. In these theories, the $10^{-35}$ m value is only the false, apparent value of the Planck length when you do experiments at scales large enough so that only four dimensions seem to exist.
If you have more than one extra dimension, you can have nontrivial topologies. The Calabi-Yau stuff is one possibility, which is purely motivated by string theory. Since string theory is probably wrong (e.g., we don't see evidence for supersymmetry at the LHC), there is no really good reason to focus on Calabi-Yau manifolds. Furthermore, results from the LHC seem to show that large extra dimensions don't exist: http://arxiv.org/abs/1012.3375
A: I think they generally refer to the dimensions of the Fiber in a Fiber Bundle.
