It is somewhat hard to understand the basic concept of a quantum computer, but I think it's worth spending a bit of time with this idea, because it will be a very important new technological development in our near future that will without doubt affect your life.
Before I try to explain what quantum computing is all about allow me to give you a real world problem that is nearly impossible to solve with classical computers but theoretically easy to compute with a quantum machine. Imagine a pizza delivery guy who has to deliver pizzas across town and he needs to find the fastest route. If there are two pizzas to deliver the number of possible solutions are only two and it's still very easy to decide which one is the shortest route by measuring both and comparing them to each other. If there are three orders the problem becomes more complex, but still manageable with eight solutions. If each address is represented by a letter the possible solutions are: ABC, ACB, BAC, BCA, CAB, CBA. At four pizzas the numbers of solutions increases to 16. At just 12 addresses the number of solutions already increases to a manually unmanageable 4,096 possible routes. This is still possible to calculate with a classic computer with a simple algorithm, but the amount of time to run the code will already take considerable time. Now imagine if we need to deliver a thousand pepperonis. The number of solutions is going to be so high (1,267,650,600,000,000,000,000,000,000,000) that it would take ages for a classical computer to solve it. By time the calculation finishes we could have the pizzas delivered taking a random inefficient route. Without a quantum computer when we face a problem like the pizza deliveries we either have to use more resources or time than it would be needed in theory to get the job done. This means we have huge inefficiencies in many of our systems. Quantum computers by promising a solution to these problems will guarantee huge optimizations of our processes. This will result in significant disruptions of our current systems and generate new wealth for humanity, just like effects of division of labor or classical computing did in the past.
So how does it work exactly? As the Wikipedia article about quantum computers explains: a classical computer has a memory made up of bits, where each bit represents either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or, crucially, any quantum superposition of these two qubit states; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8. See the pattern? The quantum computer is by default designed to deal with the kind of problems the pizza delivery guy faces. The more addresses there are the more calculations need to be done, but unlike with classical computers the amount of time required to perform the calculation doesn't increase exponentially.
Quantum computing does have a drawback though. Quantum physics is not a precise science, it is all based on probabilities. So the solution given by a quantum computer to the pizza delivery guy will not be a definite shortest route only the most likely shortest route. However when dealing with complex problems most of the time it is adequate for us to have a fast and good answer, it's not required to have the absolute certain perfect answer.
By employing quantum computers we will be able to calculate things like the weather or the outcome of government policies. I can imagine it will also be able to help solving engineering problems that have infinite number of possible solutions.
Quantum computers are specialized for certain tasks only. They are not going to replace classical computers because they are not faster in solving regular calculations. The two systems will be used together, a bit similar like the creative and rational side of our brain works together at solving problems.
While we're still years away from functioning quantum computers it is certainly not science fiction. Many organizations and corporations like the Canadian D-Wave Systems are working on it. Basic few qubit computers have already been built and plans for more complex ones are drawn. This new technology will happen in our life-time and it will change our world at least as much as classical computers did a few decades ago.