What is Poisson process, compound Poisson process?

A Poisson process with rate $\lambda > 0$ is a counting process $N_t$ with $N_0 = 0$, independent increments and $N_t - N_s \sim \mathrm{Poisson}(\lambda(t - s))$. A compound Poisson process is $J_t = \sum_{i=1}^{N_t}Y_i$, with the jump sizes $Y_i$ independent, identically distributed and independent of $N$.