A physical theory is a mathematical framework together with some rules for extracting quantities that one measures. Different physical theories often make use of very different mathematics. For example, general relavitity, which provides a classical description of gravity, is based on geometry; the quantum field theories that describe particle physics are instead built on algebra (representation theory, Hilbert spaces, etc). In recent years we have come to realize that a given physical system can sometimes have two or more equivalent theoretical descriptions that make use of very different mathematical frameworks. These different physical theories are then said to be dual to each other. Typically, however, each theory is useful in a different physical regime of the system.
One of the most dramatic examples of such a duality is the AdS/CFT correspondence. Broadly speaking, this is a duality between a theory that contains gravity, and thus in an appropriate classical limit is described by general relativity, and a quantum field theory without gravity. What is particularly striking is that these two theories live in different numbers of spatial dimensions! To make the discussion more precise, suppose that the quantum field theory depends on a parameter, that we'll call the coupling constant g. This theory is well-understood mathematically only for g very small, essentially as a power series around g=0. On the other hand, the classical limit of the dual gravitational description, which is well-understood in terms of general relativity, typically corresponds to 1/g being small. Much of the evidence for AdS/CFT relies on the existence of quantities that one can a priori argue are independent of g, and thus should give the same physical answers in both theories. This often leads to remarkable mathematical identities that must hold if the AdS/CFT duality is true; these can then be proven independently. Much of my work has been in this area, essentially building up circumstantial evidence for the conjectured duality, and trying to understand better how physical quantities in each theory should be matched.
Although originally formulated in string theory, the AdS/CFT correspondence has grown into a subject in its own right. In the last couple of years these ideas of duality have even been applied to low-energy systems, such as superconductors and other areas of condensed matter physics. This leads to the exciting possibility that such dualities might be tested experimentally in laboratories.