# Difference between a long tail and normal distribution

Wikipedia article about "long tail" says that:

A probability distribution is said to have a long tail, if a larger share of population rests within its tail than would under a normal distribution.

I am confused about this. Isn't a normal distribution exponential one? Meaning even large values of x will have some corresponding value of y, thus making those values of x "rest within its tail"?

Or does this sentence mean that area of tail in long-tail-distribution is more than are of tails in a normal-distribution?

Also what would be the difference between a long-tailed and a fat-tailed distribution?

Recall that the density function of the (standard) normal has shape $ke^{-x^2/2}$, where $k$ is a constant. If $x$ is at all large, $e^{-x^2/2}$ is very close to $0$. Compare this with even an exponential distribution, which (for the same variance) has density function $e^{-x}$ (for $x\ge 0$). Although $e^{-x}$ decays rapidly, it does so far more slowly than $e^{-x^2/2}$.
And consider now the Cauchy distribution, which has density function of shape $\frac{k}{1+x^2}$. This decays far more slowly than $e^{-x}$, and very much more slowly than $e^{-x^2/2}$: it is very much long-tailed. Or consider the distribution with density $\frac{1}{(1+x)^2}$, for $x\ge 0$. This density function decays so slowly that the mean of a random variable with this distribution does not exist (informally, is infinite): the tail wags the dog.