The shape of the normal curve depends on the mean of the distribution $\mu$ and its standard deviation $\sigma$. The distribution with $\mu =0$ and $\sigma =1$ is called the standard normal distribution.
You can transform any normally distributed variable $X$ into a standard normal variable $Z$ by subtracting the mean from every value of x and dividing by the standard deviation (resulting in a narrowing or widening of the curve).

This gives you: $z=\frac{x-\mu }{\sigma }$.

The standard normal variable $Z$ allows you to determine probabilities for any given normal variable $X$, according to the following relationship: ${\rm P}(X\le x)={\rm P}(Z\le \frac{x-\mu }{\sigma })$.
This is called standardising the normal statistic X. It is useful only if you need to calculate $\mu$ or $\sigma$.