In the classical world of macroscopic objects it is easy to classify something as a particle or as a wave.
Essentially there are two differences between these two:
1) A particle is localized in space whereas a wave is not.
2) Particles don't interfere. Waves do.
Now consider a quantum object such as an electron. If you try and detect the electron you will always find either a whole electron or no electron-never half an electron. So in this sense the electron behaves like a particle.
However the electron can exist in a superposition of states. For example it can have spin up and spin down at the same time . This is very difficult to visualize this but it turns out that when we model this mathematically the mathematics is just the Mathematics of Waves or Interference. This is what we mean by wave particle duality.
So for example in a two slit interference experiment the photon exists in a superposition of two states - State 1 where it went through the first slit and State 2 where it went through the second slit. In a way it does both at once. This is again difficult to picture but the mathematics of the interference pattern we observe is very simple.