In quantum mechanics the movement (more precisely, the state) of a particle in time is described by Schrodinger's equation, a differential equation involving a wave function, psi(x,t).
|psi(x,t) |^2 is interpreted to be the probability density of a particle moving in time; that is, the probability that the particle lies in-between x=0 and x=1 is: Integral (|psi(x,t)|^2) dx from x=0 to 1.
To answer your question, normalizing a wave function means to multiply it by a constant such that Integral (|psi(x,t)|^2) dx from -infinity to infinity = 1. In plain English, the particle has to be somewhere on x at any given time, 100% of the time. So normalizing the wave function simply ensures that |psi(x,t)|^2 can be interpreted as a probability density function.
Normalization is deeply related to quantum mechanics and quantum mechanics is a science of probability.Since the probability density i.e. probability per unit volume is to find the position of a particle at a specific point in space.
Now the question arises that
what to do with the complete space? Well we know that the probability of finding a particle in complete area is always equal to one. Similarly here in this case we say that the probability of finding the particle in the entire space is equal to one. This condition is called normalization and is represented as