What is the use of Mersenne Primes in cryptography
I can think of two places where we use a Mersenne Prime within cryptogaphy:


As a modulus within a prime elliptic curve.  $2^{521}-1$ is a prime, and so we can define an elliptic curve using $GF(2^{521}-1)$, which is in moderately common use.  One reason we use such a modulus (rather than another prime of approximately the same size) is that the special form of the prime makes computing $x \bmod 2^{521}-1$ easy.
In the (rather obscure) Carter Wegmen Counter mode, we use the fact that $2^{127}-1$ is prime; we use that prime, rather than another value, because it is approximately the correct size, and (as above) computing $x \bmod 2^{127}-1$ is easy.


In both cases, the special property that we take advantage of is that they make computing the modulo operation cheap (by using shifts and adds).