Suppose I have a probability distribution $A$ with continuous support over $\mathbb{R}$. Suppose $A$ has a sequence of finite (central) moments $\mu_1, \mu_2,\ldots,\mu_n$. I understand that $\mu_1$ is the mean, and $\mu_2$, $\mu_3$ and $\mu_4$ define variance, skewness, and kurtosis of the distribution, respectively.
I am wondering about the meaning of $\mu_5, \mu_6, \mu_7,\ldots$ When they are finite, what do they represent about the distribution $A$?
I understand that the odd central moments of the symmetric distribution are zero, so I am assuming that odd moments are related to the skew. What do even higher moments represent? I am particularly curious about $\mu_6$.
Also, suppose all moments of $A$ are finite. What does that say about $A$? Does it mean that $A$ has a specific representation? I've heard somewhere that all finite moments of $A$ with support $\mathbb{R}$ means that the tails of $A$ decay exponentially. Is that true? If so, can someone point me to a proof?