P(X = k) = lambda^k e^(-lambda) / k!. Mean and variance are both lambda, which is its signature and a standard interview check.
Arises as the limit of a binomial with many trials and small success probability - the distribution of rare events among many opportunities. Order arrivals, defects and rare failures are commonly modelled this way.
The dispersion test. Because mean equals variance, comparing the two in real data tells you whether the Poisson assumption holds. Financial event counts are usually overdispersed - variance exceeds the mean, because arrivals cluster rather than being independent.