In 1960, physicist Eugene Wigner published the now-famous article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." It claimed that math concepts often apply to contexts in physics quite different from their original discovery or development.
Wigner writes: "It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena." That is, in an unexpected way, the mathematical structure of a physics theory often suggests ways to further advance that theory....thereby reflecting "some larger and deeper truth about both mathematics and physics."
As a first step, read Wigner's paper, which I found to be interesting but not a trivial read. Then, browse Wikipedia's summation of the paper and informed responses.
Recently (2008), Robbert Dijkgraaf wrote an updated commentary on Wigner's thesis. It is labeled Mathematical Connections.
Finally, Alex Kasman has complied an interesting list of fiction devoted to the theme of Unreasonable Effectiveness. It will give you a different perspective on the notions of effective and useful in terms of mathematics.