A binomial experiment is a series of n independent trials, each with two possible outcomes (often labeled "success" and "failure"), with a constant probability, p, of success.  The total number of successes, k, is given by the binomial distribution.

If there are more than two possible outcomes, use the multinomial distribution.

If the sampling is without replacement, use the hypergeometric distribution

For large n, and p not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with mean μ = n p, and variance = n p (1-p).  For large n, and p close to 0, it can be approximated by a Poisson distribution with parameter λ = n p. As n->infinity these are the limiting distributions (providing p=constant in the normal case, and p->0, np=constant in the Poisson case).