For example by compactifying one of the know string theories on a 6 dimensional space one finds many different 4 dimensional theories with various physical features. Although the concept of duality goes back to the last century, it has only been the last few years that it string theories were seen to possess a duality structure. Simply put, two theories are said to be dual to each other if they describe the same physics using different language. Because we usually only have perturbation theory to investigate a theory with, two theories may appear completely different. In this way it is thought that all the string theories are in fact different aspects of one underlying theory, M theory .
Duality is thus a very powerful tool in understanding the full physics of a theory.
The standard model is built out of a type of theory known as a gauge theory. A gauge theory arises when one imposes that a particular symmetry of a theory is valid even when the parameter labelling the symmetry is allowed to vary over spacetime. In quantum theory for example it is well known that you may change the phase of the wave functions by an arbitrary amount without altering any the physical content or structure of the theory, provided that you change the all wave functions in the same way, everywhere in space. To gauge this symmetry one lets this phase change vary over spacetime. In order to maintain this as a symmetry of the theory it turns out that you must introduce the electromagnetic force.