This is a pretty open-ended question, but I thought I may ask. The normal distribution that is generally used is taken on a variation of the equation: $$y=e^{-x^2} $$
But I've noticed that there are quite a few other functions that behave like this: $$y=\frac{1}{1+x^2}$$ $$y=\operatorname{sech}(x)$$ $$y=\arctan(\frac{1}{x^2})$$
It's pretty apparent that most of them have some form of $x^2$ in them, and I know the $\operatorname{sech}(x)$ actually has a name (hyperbolic secant distribution). What other functions behave like this? And can they all be used the same way as the normal distribution?
Many curves are "bell shaped". The ones you mention are examples.
But the normal density goes to zero (as $x \to \infty$) at a certain rate, not shared by some other bell shaped curves.
$\arctan(1/x^2)$ and $1/(1+x^2)$ go to zero only at the rate $O(1/x^2)$, which is much slower than $\exp(-x^2)$.
$\mathrm{sech}(x)$ goes to zero at the rate $O(\exp(-x))$ which, although much faster than $O(1/x^2)$, is still much slower than $\exp(-x^2)$.
Can these others be used in the same way as the normal density? No, only the normal density works in the Central Limit Theorem.