Let $\varphi$ be the characteristic function of an infinite divisible distribution. It can be expressed in the form $\varphi = e^\psi$ with
$$\psi(\lambda) = i \lambda a - \frac{\sigma^2 \lambda^2}{2} + \int_{\mathbb R} \left(\exp(i \lambda x) - 1 - \frac{i\lambda x}{1+x ^2}\right)\frac{1+x^2}{x^2} \nu(dx)$$
where $(a,\sigma^2,\nu)$ is the Lévy-Khintchine triplet.
What is the intuitive meaning of this triplet?
For example, how can one understand that $\sigma^2=0$ for the compound Poisson distribution?