"Physical" meaning of higher moments (their values and their existence) 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?

For your final statement the classic distribution to consider is the log-normal distribution.    It is an example of a case where all moments are finite, but the moment generating function does not exist and the moments do not determine the distribution.
Take for example the log-normal distribution with parameters $\mu=0$ and $\sigma = 1/\sqrt2$ which has a distribution on positive values with density $$f(x)=\frac{1}{ x^{1+\log_e x}\sqrt{\pi} }$$ 
This has finite moments about $0$ of $E[X^n]= \exp(n^2/4)$ but the density of the tail does not decay exponentially. 
