Quantum Mechanics is the part of physics based on the discovery that things tend to get fuzzy as they get small. I doubt that I should try to explain the Schrödinger Equation here (it is a differential equation, which explains how the wave-function of a particle is affected by the potential energy distribution it faces; the frequency of the wave determines the particle's momentum, and its amplitude the probability the particle is at any point; it corresponds to the classical equations which show how a particle will move in response to forces).
Quantum Mechanics so far has been found to relate to cryptography in two different ways.
A quantum computer might be able to operate like the ultimate parallel computer: it might be able to solve a problem for every possible set of initial conditions at once! However, it could only give us one of the solutions when it finishes.
Quantum cryptography is the public-key cryptography answer to invisible ink; one can send two particles out from a central location to two communicating parties, allowing both parties to co-operate in generating a random one-time pad common to both of them that no one else could have intercepted.