The first thing you need to get to grips with is that particles are waves. This can be shown with a simple experiment called the double slit experiment, which I will attempt to explain.
Imagine a water wave travelling across a tank. Then imagine you place a wall in the middle of the tank, and place two thin slits in it. If you create a wave (by dropping a stone etc) on one side of the wall, it will travel through the two slits and interfere like this.
The double slit experiment does the same thing, but for light. If you have a wall with two slits in it and shine a beam of light through the slits onto a flat screen behind, you can see a similar interference pattern on the screen.
This shows that light acts as a wave.
Now imagine that rather than a beam of light you can create a steady stream of electrons. Electrons are a small "fundamental" particle ("fundamental" means they cannot be broken down into smaller components). If you point your electron stream at your two slits you will see a very similar interference pattern as before! Until this experiment was done it was believed that electrons were solid particles (like billiard balls), but this showed that they also act as a wave!
Since we have shown that particles can also show wave-like properties, can we show that waves can have particle-like properties? It was shown by Einstein and Arthur Compton that light can in fact be shown to be made up of particles, due to the fact that light must have momentum.
This is known as wave-particle duality.
As I said at the beginning, waves and particles are the same thing. There are some "waves" like electromagnetic waves which make particles move. These are only called "waves" because it is easier to model and calculate that way. It is possible to describe the interaction as 2 (or more) particles (but it is considerably more difficult).
I hope this answers your question.