In general
The cumulants of a random variable \(X\) are defined by the cumulant generating function, which is the natural log of the moment generating function:
\[\as{ K(t) &= \log M(t) \\ &= \log \Ex e^{tX}. }\]The \(n\)-th cumulant is then defined by the \(n\)-th derivative of \(K(t)\) evaluated at zero, \(K^{(n)}(0)\).
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the third central moment, but fourth and higher-order cumulants are not equal to central moments.
Exponential families
The function \(\psi(\theta)\) in an exponential family is often referred to as its cumulant generating function, although technically the cumulant generating function would be
\[K(t) = \psi(t + \theta) - \psi(\theta).\]Typically, however, the distinction is unimportant, as derivatives still yield the appropriate cumulants:
\[\psi^{(n)}(\theta) = K^{(n)}(0).\]