A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: 
$$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$

or its characteristic function. 

As recalled in this question it is also the only distribution for which the sample mean and variance are independent. 

What are other surprising alternative characterization of Gaussian measures that you know ? I will accept the most surprising answer 

My personal most surprising is the one about the sample mean and variance, but here is another (maybe) surprising characterization: if $X$ and $Y$ are IID with finite variance with $X+Y$ and $X-Y$ independent, then $X$ and $Y$ are normal.

Intuitively, we can usually identify when variables are not independent with a scatterplot.  So imagine a scatterplot of $(X,Y)$ pairs that looks independent.   Now rotate by 45 degrees and look again: if it still looks independent, then the $X$ and $Y$ coordinates individually must be normal (this is all speaking loosely, of course).

To see why the intuitive bit works, take a look at

$$
\left[
\begin{array}{cc}
\cos45^{\circ} & -\sin45^{\circ} \newline 
\sin45^{\circ} & \cos45^{\circ} 
\end{array}
\right]
\left[
\begin{array}{c}
x \newline
y
\end{array}
\right]= \frac{1}{\sqrt{2}}
\left[
\begin{array}{c}
x-y \newline
x+y
\end{array}
\right]
$$
