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.