1. The Photon = In Thomas Young´s Experiment, light was sent through double slits and produced a wave pattern. However, the Albert Einstein´s Photoelectric effect indicates that light behaves like a particle. So you really need need a theory that incorporates both views (wave and particle) to be a complete theory for the photon. (BTW, there is a video on YouTube that shows how you can have fun with the photoelectric effect.)
2. The Electron = Robert Millikan´s Oil Drop Experiment showed that the electron was a particle. (A simpler explanation is with this YouTube video.) However, the Davidson-Germer Experiment showed that the electron also was a wave. Thus, again, a complete theory must account for both the wave and particle nature of the electron.
3. The Atom = John Dalton summarized something that all chemists had already observed in his atom theory of chemicals, thus establishing the particle nature of atoms. His idea of atoms eventually led Lothar Meyer and Dmitri Mendeleev to come up with the periodic table of elements (an interactive one is here). However, Otto Stern and Walter Gerlach demonstrated that even neutral atoms follow wave principles and discovered the spin characteristic of atoms. (A simpler explanation of the Stern-Gerlach experiment is available on YouTube.) Once again, a complete theory must account for both the wave and particle natures of the atom.
It was Louis deBoglie that formalized that any particle in motion must have a wave associated with it that obeys the relation where and h=Planck´s constant, and momentum and the wave vector where with wavelength. This even works for the photon, since the relation has been experimentally verified (which also proves that even though photons have energy and momentum, they have no mass).
Today, as a result of one of the four axioms of QM, the operators are substituted for the variables in particle theory in the following way (source = »Modern Physics And Quantum Mechanics« by Prof Elmer Anderson; ISBN=0-7216-1220-2; copyright 1971 by WS Saunders Company; page 149):
In the position domain (along x-axis):
In the momentum domain (along x-axis):
This ensures that you have both particles and waves embedded into one theory that explains all behavior of particles (including photons). The theory has held up very well under all experiments. There are things QM does not explain, but the things it does explain, it explains very well. It should be noted that in QM, E is invariant between the position and momentum domains (cf above). So energy is perhaps the most important quantity in QM.
I´ve skipped a lot of detail of QM, but I hope to have elucidated wave-particle duality a bit and hope this helps.