The literal meaning of the word “dual” implies two but in philosophy “dual” usually means “not one.” In physics dual means equivalent. There are multiple descriptions of physical reality. These descriptions are not necessarily equivalent but if they are then we call those dual theories.
The concept of duality in physics can be best described by dual universes. We do not know whether dual universes actually exist or not, but we can think of them as mathematical universes.
Physical laws are expressed in terms of equations. What is an equation? It has a left hand side and a right hand side. The equation of a physical law maps the physical quantities on the right hand side to the physical quantities on the left hand side. An equation expresses a functional relationship.
If there is a dual universe every physical quantity in this universe would have a corresponding physical quantity in the dual universe in such a way that the mathematical equation of the physical law is the same in both universes. This is the key point. After the transition to the dual universe the mathematical form of the physical law remains the same but the physical quantities are replaced by their duals.
Transition to the dual universe is known as duality transformation which transforms all physical quantities into their duals. Duality transformations typically involve simple rotations but these duality rotations can be very abstract too.
I mentioned earlier that in physics dual means equivalent. The equivalence refers to the equivalence of the form. Dual theories are equivalent because their equations have the same form. The physical quantities, however, are different. For example, an electron in this universe may appear as a magnetic monopole in the dual universe. In both universes the physical law is expressed using the Maxwell equations. The equations are exactly the same but the physical quantities have been switched with their duals. This is the idea behind the unification through a duality transformation.
If an electron in this universe appears as a magnetic monopole in the dual universe and if I know the details of the duality transformation then I know something about the nature of the real entity (unified field or particle) that exists, perhaps, in a higher dimensional realm. Electron is one projection and magnetic monopole is another projection of the same entity in the dual universes.
In this type of unification the equation itself is taken as the fundamental entity. The physical quantities are mere projections. The implication is that the abstract realm of the equation is more fundamental than the physical realm.
Discovering the laws of physics and writing them down as mathematical equations is one thing and finding the solutions of those equations is another. It took Einstein a decade to formulate the law of gravitation in terms of an equation. It took a century and hundreds of physicists to find all the mathematical solutions of that equation in different physical settings. People are still solving the Einstein equation. Same goes for the Dirac equation. It took Dirac only a year to come up with his famous equation for the relativistic motion of an electron. People are still applying Dirac equation and finding solutions in the presence of various external fields. Finding solutions can be more difficult than formulating a law of physics in terms of an equation.
Under the duality transformation the equation remains the same while the physical quantities are exchanged by their duals. This means that we understand both universes in terms of the same law. If it is difficult to solve the equation for a particular physical quantity we then switch to the dual universe and solve the same equation for the dual quantity then convert that solution to it equivalent in this universe. Finding a solution for the dual quantity is sometimes easier.