Quantum mechanics describes reality as dynamics of abstract vectors in complex vector space with defined norm, which is called by mathematicians, Hilbert space. Every thing you can measure, called observables, are functions acting on elements of this space - vectors. Such functions acting on vectors in linear way, so we call them operators. Given vector, describing some physical state of the system, operator acting on such vector gives you another vector, transforming oryginal one. In some way it looks like rotation.
If you want to know what is the result of the measure, you have to choose observable, the craft operator which gives you measurement results, and act operation on vector of the physical state of the system. Then you will get result - a vector describing system after measurement.
But what about values of measured things? There's any value. There's only another vector. Why? Because typical ordinary quantum system does not have any certain value of any observable!
But you may say, some of the vectors are special. If you are able to prepare quantum system in a state of that particular vector, some observable, let's say velocity, may have certain value. Other observables usually not, but this one, and any related to it, like momentum, would have.
So for typical quantum system you cannot get any certain value, but for some artificially crafted, you will.
As vector space usually has property it have basis, you can use such crafted vectors as your basis. Typical space of quantum system has infinite dimensionality, because there's a lot of possible velocities. But it is just a technical problem.
In that case you may do the following: you start with your system, described by some, abstract vector in this vast space. But now you are able to decompose this vector into sum of elements of your basis. Remember, every basis vector has defined velocity ( and undefined rest of possible observables, but you do not care for a moment, because you want to know velocity). So sum you would get is large. In typical case it contains many basis vectors with different velocities, every of it with some coefficient. Then you act observable operator on it, one by one, and after that, some coefficients will change. Then you take a look and see, some velocities have bigger coefficients than others. You can check it just performing normal projection in direction of this or that basics vector.
You still don't know what velocity your system have!
But is responsible to expect that velocities with bigger coefficients will dominate, and in fact those coefficients measure probability of your measurement gain.
But then you changed your mind, and want to know position instead.
Oh, it is just simple: you have to change your vector space basis. Instead using vectors with defined velocity, you would like to use another ones with defined position. Well, as everything comes at price, single vector with defined velocity, usually have to be defined as infinite sum of vectors with definite position. And vice versa.
In fact everything you do in quantum mechanics is just exchange one vector space basis with another and checking of coefficients of such decompositions.
Duality? This is just fancy name for such vector space basis equivalence.
