The binomial distribution is used when you are interested in the probability of the number of successes n in N trials when the probability of success on a single trial is p.

You use the Poisson for the number of occurences of something when the mean number is known (its like the probability of n cells in a sample when the mean number of cells in a sample is N - with side conditions something like: the presense of one cell does not influence the presense or absence of any others) Another example would be the number of customers arriving to be served in an hour, ..

CB