If you have always wanted to know how a quantum computer works, then you might get a tiny flavour from this animation ...
but in all probability you are just going to be mystified. Quantum computers are difficult to understand because they rely on the mathematics of quantum mechanics and most people don't understand the math.
This animation, which accompanies an interview with two experts on quantum computation, might give you some idea as to why quantum computers are more powerful - or potentially more powerful - than a classical computer.
It is too easy to say that the reason a quantum computer is more powerful is that a qubit, or quantum bit, can represent a zero and a one at the same time. This seems like a powerful idea, but it doesn't really give you much that is new in terms of computation. It is only when you allow a set of qubits to be entangled do you get really new behavior. When qubits are entangled the result of one measurement affects another and you can use it for encryption and computation.
The big problem is that entangled states are corrupted by any interactions with the outside world - a problem known as decoherence. So far this has made building quantum computers with more than a small number of qubits difficult.
Exactly how all this works is difficult to comprehend, but despite the promise of computation it is important to realize that a quantum computer cannot compute anything that a classical computer cannot. Indeed the operation of a quantum computer can be simulated by a classical computer, but it might take longer than the lifetime of the universe to complete the job. Quantum computers promise fast solutions nothing more. Even so this is enough of a promise.
If you have heard about D-Wave's multi-qubit quantum computer then it is worth mentioning that this is a different sort of quantum computer. It is a quantum annealing device which can be used to solve specific optimization problems. It is more like a quantum analog computer than anything else. Even so, Google recently bought one to help with its research into AI.
A quantum computer would be very handy. However, there is a doubt at the back of every theoretician's mind that it might not be possible to build such a thing. There might be some "no-go" theorem which restricts what can be done so as to rule out the existence of a quantum computer.