When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect of the waves is the sum of the individual waves' effects. However, it does not imply that the individual waves affect the propagation of other waves.
Interference occurs in mechanical waves, such as sound waves, waves on a string, and surface water waves. Mechanical waves correspond to the physical displacement of particles. Hence, interference implies that the physical displacement of a medium's particles is given by the algebraic sum of the displacements that the individual waves cause. Interference also occurs in other waves like electromagnetic waves, in which case the electric and magnetic fields add up vectorially.
However, the principle of superposition does not apply to certain waves. That is, the resultant wave may not be a simple algebraic sum of the individual waves. In this case, the effect is nonlinear, complicating the description of the phenomenon. These waves are said to be nonlinear waves.
Alternative bands of constructive and destructive interference may result in interference patterns, a smoking gun signature of wave phenomenon occurring in nature.
This text is adapted from Openstax, University Physics Volume 1, Section 16.5: Interference of Waves.
