Discrete distributions come from a variety of backgrounds, but perhaps the most common relate back to the simple Bernoulli trial, which chooses between two outcomes, called success and failure here, whether you count the number of successes, the number of failures until first success, the number of failures until n success, and so on. Other important origins are urn models with different sampling and replacement scenarios. Others are limit distributions of various kinds of processes.
BernoulliDistribution — distribution of an outcome of a Bernoulli trial
BinomialDistribution — the number of successes in Bernoulli trials
GeometricDistribution — the number of failures before the first success
NegativeBinomialDistribution — the number of failures before the success
PascalDistribution — the number of trials for the success
HypergeometricDistribution — sampling without replacement until balls are drawn