Well, currently we encode data as 0 or 1. Voltage or no voltage. We can then use the electrical properties of the components to calculate the result of the previous input. Mostly, this means using logic to get the answer. For example, using an adder circuit to add 2 numbers together.
Now enter quantum states of particles. These particles have spin, energy levels etc. This means that there are a number of different ways to encode the information into the particles (this is roughly equivalent with voltage in this respect). This would drastically reduce the size of 1 bit, meaning that we can put more of those on a given surface. However, the nature of the quantum world needs a completely different approach to computing, one which promises to be superior to the way things are done at the moment. The major problem is in how to evolve the quantum state of a system in such a way that the correct answer comes out, but unfortunately, I don't know anything about that either. Maybe they use magic :P