You can simulate quantum processes on a classical computer by keeping track of the amplitude of every basis state in the tensor product space of the thing you are simulating. Basically, you're running a physics simulation. It's extremely inefficient, and in terms of classical RAM required it is exponential in the number of qubits you are trying to simulate, but useful for prototyping, debugging, and learning about quantum algorithms.

Google wrote a nice one that makes use of WebGL: http://www.quantumplayground.net/