How to imagine higher dimensions?
The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime).
Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius $R$ is compact and two-dimensional - every point on it can be described by two angles, and it's volume is finite as $\frac{4}{3}\pi r^2$. Ordinary Euclidean space $\mathbb{R}^3$ is non-compact and three-dimensional - every point in it is described by three real numbers (directed distance from an arbitrarily chosen origin), and you can't associate a finite volume to it.
Note that, on the sphere, you can keep increasing any one of the coordinates and, sooner or later, you will return to the point you started from. All dimensions here are "small"/compact. In Euclidean space, you never return to the origin, no matter how far you go. All dimensions are "big"/non-compact.
An infinitely long cylinder is now an example of where the two dimensions are different. Take as coordinates the obvious two - the length (how far "down"/"up" on the cylinder you are), and the angle (where on the circle that's at that length you are). The length dimension is non-compact - you never return to your starting point if you just keep increasing that coordinate. The angle coordinate is compact - you return after $2\pi$ to your starting point, and the "size" of the dimension is the radius of the circle. This is an example of a "curled up dimension". If you are far larger than the radius, you might not even notice you are on a cylinder, and instead think you are on a one-dimensional line!
1The mathematical definition is by covering properties which are not as easily translated into intuition.
In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2-dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higher-dimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is technically known as an "embedding space", but the mathematics of differential geometry allows mathematicians and physicists to describe the curvature of surfaces in purely "intrinsic" terms without the need for any embedding space, rather than in "extrinsic" terms where the surface is described by its coordinates in a higher-dimensional space--see the "Intrinsic versus extrinsic" section of the differential geometry wiki page. And all this has a practical relevance to physicists, since Einstein's theory of general relativity uses differential geometry to explain gravitation in terms of matter and energy causing spacetime to become curved (see here for a short conceptual introduction to how spacetime curvature can explain the way particle trajectories are affected by gravity).
With these ideas in mind, if you want to understand Greene's comment about higher dimensions being "curled up", picture the surface of a long cylinder or tube, like a garden hose. This surface is 2-dimensional, but you only have to travel a short distance in one direction to make a circle and return to your place of origin--that's the "curled up" dimension--while the perpendicular direction can be arbitrarily long, perhaps infinite. You could imagine 2-dimensional beings that live on this surface, like those in the famous book Flatland that has introduced many people to the idea of spaces with different numbers of dimensions (and there's also a "sequel" by another author titled Sphereland which introduces the idea that a 2D universe could actually be curved). But if the circumference of the cylinder was very short--shorter even than the radius of atoms in this universe--then at large scales this universe could be indistinguishable from a 1-dimensional universe (like the "Lineland" that the characters in Flatland pay a visit to). So a similar idea is hypothesized in string theory to account for the fact that we only experience our space as 3-dimensional even though the mathematics of string theory requires more spatial dimensions--the extra dimensions are "curled up" into small shapes known as Calabi-Yau manifolds, which play a role analogous to the circular cross-sections of the 2D cylinder or tube I described (although in brane theory, an extension of string theory, it's possible that one or more extra dimensions may be "large" and non-curled, but particles and forces except for gravity are confined to move in a 3-dimensional "brane" sitting in this higher-dimensional space, which is termed the "bulk").