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:
\[\begin{align*} K(t) &= \log M(t) \\ &= \log \Ex e^{tX}. \end{align*}\]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:
\[\psi^{(n)}(\theta) = K^{(n)}(0).\]