Since any given particle or system of particles is a wave, it can be represented by a wave function. This wave function describes the state of the particle/system and can be used to determine properties of the system such as position and momentum. Of course, in accordance with the uncertainty principle, this wave function won’t give you the exact momentum and position of a particle but rather the probability that the position/momentum is within a certain interval. One property of waves not yet mentioned is that they have boundary conditions. If you create a wave on a string that is fastened at both ends, for instance, then the equation for that wave must be zero at both ends of the string. This feature leads to quantization which means that the system can only exist in a discrete (rather than continuous) set of states satisfying the boundary conditions. In the case of the string, the wave can exist in any of an infinite number of discrete states- a state with no nodes, a state with one node, a state with two nodes, etc.
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