We want a density that is symmetric about some center value with some spread.
Let $f: \R \to \R $ be a density.
If there exists $\mu \in \R $ and $\sigma
\in \R $ with $\sigma > 0$ so that for
each $x \in \R $
\[
f(x) = \normaldensity{x}{\mu }{\sigma }
\]
We call the special case when $\mu = 0$ and $\sigma = 1$ the standard normal density or standard gaussian density.
The maximum of a normal density is $\mu $.