The simplest way to hide extra dimensions from view is to imagine that they are "compactified"—curled up into a tiny ball (or other geometrical configuration) with an extent much smaller than what can be probed by current experimental apparatus. In the 1990s, however, a new possibility arose, as scientists came to appreciate the role of "branes" in higher-dimensional physics. A brane, generalizing the concept of a membrane, is simply an extended object: A string is a one-dimensional brane, a membrane is a two-dimensional brane, and so on, up to however many dimensions may exist. A remarkable feature of such objects is that particles may be confined to them, unable to escape into the surrounding space. We can therefore imagine that our visible world is a three-dimensional brane, embedded in a larger universe into which we simply can't reach.That's from a review of Lisa Randall's new book, Warped Passages.
Gravity, as the curvature of spacetime itself, is the one force that is hard to confine to a brane; the extra dimensions must therefore have some feature that prevents gravity from appearing higher-dimensional. (For example, in four spatial dimensions, the gravitational force would fall off as the distance cubed, rather than the distance squared.) One possibility, proposed by Nima Arkani-Hamed, Savas Dimopoulos and Georgi ("Gia") Dvali, is that the extra dimensions curl up into a ball that is small without being too small—perhaps as large as a millimeter across in each direction. Randall, in collaboration with Raman Sundrum, showed that an extra dimension could be infinitely big, if the higher-dimensional space was appropriately "warped" (hence the title of her book).