When adding two independent random variables $X$ and $Y$, their respective moment generating functions $E\{\exp(t\,X))\}$ and $E\{\exp(t\,Y)\}$ are multiplied to yield the moment generating function of the sum $E\{\exp\bigl(t\,(X+Y)\bigr)\}$. This results in a convolution of the respective probability density functions.
If you take the logarithm of the moment generating functions (of the random variables and their sum), you get the cumulant generating functions that are added for the cumulant generating function of the sum of the independent variables.
Developing this function into a power series yields cumulants as successive coefficients.
The constant term is 0, the linear coefficient is the mean of the respective distributions (means of independent variables add when adding the variables), the quadratic coefficient is the variance, the next two terms are called skew and kurtosis. However, distributions for which all cumulants except mean and variance are zero are normal distributions.
The only cumulant guaranteed to be non-negative in any probability distribution is the variance: that is a significant contributor to the central limit theorem since variances cannot cancel each other.