An invention is useful if it provides some practical benefit, or helps people complete real world tasks. However, patents may be granted for inventions even if their use is merely humorous, such as a musical condom or a motorized spaghetti fork.
To fulfill this requirement, the invention must work -- at least in theory. Thus, a new approach to manufacturing superconducting materials may qualify for a patent if it has a sound theoretical basis -- even if it hasn't yet been shown to work in practice. But a new drug that has no theoretical basis and which has not yet been tested will not qualify for a patent.
Only a utility patent requires an invention to be useful: To qualify for a design or plant patent -- the other two types of patents obtained in the U.S. -- the inventor need not show utility.
To learn more about qualifying for a patent, see Nolo's Patents fsor Beginners, by David Pressman and Richard Stim (Nolo).