Determining if a number is prime or not (primality test) is an excellent example.
The naive method just check if n mod x where x = 2..square root(n) is zero for at least one x. This method can get really slow for very large prime numbers and it is not feasible to use in cryptography.
On the other hand there are a couple of probability or fast deterministic tests. These are too complicated to explain here but you might want to check the relevant Wikipedia article on the subject for more information: http://en.wikipedia.org/wiki/Primality_test
