The above formula indicates the probability of r successes out of n independent trials, where each success has probability p of occurring.
The formula can be thought of in the following way. The probability of r successes occuring, each with probability p is pr. This leaves n-r failures, and each failure has probability 1-p. We multiply all of these probabilities together because the trials are independent, and this gives us pr(1-p)n-r.
Most calculations involving the binomial distribution will be carried out with software or tables. However, it is important to realize that the values in the table do not fall out of the sky. Only a few mathematical ideas from probability and combinatorics are needed to derive tables of the binomial probability distribution.