Primes are divisible only by themselves and 1. Apart from 2, all primes are odd. Therefore, the smallest possible difference between two primes is two (not counting the primes 2 and 3 with a difference of 1). Because every third odd number is divisible by three, it is not possible to have three primes with a difference of two. But it is possible to have three primes where the difference between the least and greatest is six. Three such primes are called prime triplets. An example is 11, 13 and 17.
For further information, go to Wolfram MathWorld, for example.