The $k$th (noncentral) moment of the random variable $X$ (if it exists) is $\mu^\prime_k = E(X^k).$
How MGF generate moments. I have frequently found it useful to write the 'moment generating function' (MGF) of a discrete random variable $X$ (if it exists) is
$$M_X(t) = E(e^{tX}) = \sum_x e^{tx}p(x),$$
where the sum is taken over all values $x$ in the support of $X$.
By a Taylor expansion
$$e^{tx} = 1 + tx + \frac{(tx)^2}{2!} + \frac{(tx)^3}{3!} + \dots,$$
so we can write
\begin{eqnarray}
M_X(t) &=& \sum_x p(x) + t\sum_x xp(x) + \frac{t^2}{2!}\sum_x x^2 p(x) + \frac{t^3}{3!}\sum_x x^3 p(x) + \dots \\
&=& 1 + t + t\mu^\prime_1 + \frac{t^2}{2!}\mu^\prime_2 + \frac{t^3}{3!}\mu^\prime_3 + \frac{t^4}{4!}\mu^\prime_4 + \dots,
\end{eqnarray}
provided it is OK to change the order of the summation.
Moment generating functions get their name because it is possible
to obtain the $k$th (noncentral) moment as
$$\mu^\prime_k = M^{[k]}_X(0),$$
where the superscript $[k]$ denotes the $k$th derivative.
The second line of the equation just above shows why. Taking the $k$th derivative "gets
rid" of the first $k-1$ terms; the $k$th term becomes $\mu^\prime_k;$
and evaluation at $t = 0$ "gets rid" of all the terms after the $k$th (which still have factors of $t$). It may be instructive to
evaluate $M^\prime_X(0),\, M^{\prime\prime}_X(0)$ and $M^{[3]}_X(0)$ to see how this works.
[Note: This method assumes it is OK to exchange the order of
infinite summation and taking the derivative. This can be
justified if the MGF exists.]
Similar equations can be written for the MGF of a continuous
random variable, using integrals instead of sums. In order for the
MGF to exist, the relevant sum or integral must converge for
$t \in (-\epsilon,\epsilon)$ for some $\epsilon > 0.$ There are
a some distributions for which the MGF does not exist, including
some practical ones. Student's t distribution is one example.
In some parts of mathematics and physics MGFs are called 'Laplace
transforms' and used for topics beyond probability distributions.
Some additional uses of MGFS. Besides 'generating moments' MGFs are useful for some proofs
in mathematical statistics, including limit theorems, and
convolutions (distributions of sums of certain independent random variables).
If one knows the PDF of a distribution, it is straightforward
to find the MGF. However, there is no systematic way to get the
PDF from the MGF. For probability students, this means that it is
a good idea to memorize the forms of MGF for a variety of
frequently-used distributions.
There is a limited uniqueness property for MGFs. No two
fundamentally different distributions have the same MGF.
Sometimes 'cumulants' and 'characteristic functions' (Fourier
transforms) are used for similar purposes. But this is not
the place to do more than mention them.
