Percolation is used in almost every area of science as the simplest model for spatial disorder. The model amounts to randomly coloring each of N sites in a graph (e.g. the hypercubic lattice) either black or white with probability p and then identifying "clusters" of adjacent black sites. This very simple coin-flipping game has a rather nontrivial "phase transition" in the limit N -> oo which is apparent in the expected size of the largest cluster S: For p less than some critical value p_c, the largest cluster is negligably small, E[S] = O(log N), while for p > p_c it occupies a nonzero fraction of the system E[S] = O(N). (At p = p_c it is a fractal.) Although this scaling of the mean is well-known, however, the scaling of higher moments and the limiting shapes of the probability distribution of S are not. Here, these quantities are derived analytically and checked numerically on square lattices of up to 30 million sites. A "renormalization-group" picture of percolation is proposed that draws on classical ideas from probability theory as well as the modern theory of critical phenomena.