Most of the time, the 'weight' flows back and forth between a and b according to certain equations over time. When you measure the system- that is, when the bit interacts with the outside world, hopefully your measuring apparatus- you see a 1 or a 0, with probabilities |a^2| and |b^2| respectively.
So what you can do is get a whole bunch of these quantum bits- qubits together, and set things up so that the time-evolution of their quantum state is correlated and probabilistically moves towards something you're interested in. Say you can set things up so the bit array- which, at first, will give you a mere perfectly random bit string on measurement- becomes more and more likely to give you, say, a prime factor, or the answer to some other question.
So yes, the quantum phenomenon is that the bits of the computer are quantum objects as opposed to classical.
Entropy sucks.