Poisson is a discrete variable, but it is not finite because it can take on any value -- it does not have a pre-defined maximum, like the Binomial. However, in practice, the values of large events are very low, so the odds of seeing a very large number are very low.
S1 Definition of a Poisson Random Variable
S2: Event Counts are often Poisson
S3: Calculations of Poisson Probabilities
S4: Failure of Poisson Assumptions