The normal distribution with mean $\mu$ and standard deviation $\sigma$ has distribution function $$f_{\mu,\sigma}(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/(2\sigma^2)}.$$ To generate the standard normal, we set $\mu=0$ and $\sigma=1$ to get $$f_{0,1}(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}.$$ The parameters are chosen specifically to make $$\int_{-\infty}^{\infty} f_{\mu,\sigma}(x) dx = 1$$ for all $\mu$ and $\sigma$. You can fiddle with these parameters below to see that the area below the curve remains constant as they change.

Std-dev $\sigma$:

The shaded area represents all outcomes within one standard deviation of the mean, which is always about $68\%$.