I took a class on Quantum computing, and studied many specific QC algorithms, so I know a little bit about them.
Quantum Computers are not super-computers. On a bit-for-bit (or qubit-for-qubit) scale, they're not necessarily faster than regular computers, they just process info differently. Since information is stored in a quantum "superposition" of states, as opposed to a deterministic state like regular computers, the qubits exhibit quantum interference when mixed with other qubits. Typically, your qubit starts in 50% '0' and 50% '1', and thus when you measure it, you get a 50% chance of it being one or the other (and then it assumes that state). But if you don't measure, and push it through quantum circuits allowing them to interact with other qubits, you get the quantum phases to interfere and cancel out. If you are damned smart (as I realized you have to be, to design QC algorithms), you can figure out creative ways to encode your problem into qubits, and use the interference to cancel out the information you don't want, and leave the information you do want.
For instance, some calculations will start with the 50/50 qubit above, and end with 99% '0' and 1% '1' at the end of the calculation, or vice versa, depending on the answer. Then you've got a 99% chance of getting the right answer. If you run the calculation twice, you have a 99.99% chance of measuring the correct answer. However, the details of these circuits which perform quantum algorithms are extremely non-intuitive to most people, even those who study it. I found it to require an amazing degree of creativity, to figure out how leverage quantum interference constructively.
But what does this get us? Well it turns out that quantum computers can run anything a classical computer can do, and such algorithms can be written identically if you really wanted to, but doing so gets the same results as the classical computer (i.e. same order of growth). But, the smart people who have been publishing papers about this for the past 20 years have been finding new ways to combine qubits, to take advantage of nature of certain problems (usually deep, pure-math concepts), to achieve better orders of growth than possible on a classical computer. For instance, factoring large numbers is difficult on classical computers, which is why RSA/PGP/GPG/PKI/SSL is secure. It's order of growth is e^( n^(1/3) ). It's not quite exponential, but it's still prohibitive. It turns out that Shor figured out how to get it to n^2 on a quantum computer (which is the same order of growth as decrypting with the private key on a classical computer!). Strangely, trying to guess someone's encryption key, normally O(n) on classical computers (where n is the number of possible keys encryption keys) it's only O(sqrt(n)) on QCs using Grover's algorithm. Weird (but sqrt(n) is still usually too big).
There's a vast number of other problems for which efficient quantum algorithms have been found. Unfortunately, a lot of these problems aren't particularly useful in real life (besides to the curious pure-mathematician). A lot of them are better, but not phenomenal. Like verifying that two sparse matrices were mulitplied correctly has order of growth n^(7/3) on a classical computer, n^(5/3) on a quantum computer. You can find a pretty extensive list by googling "quantum algorithm zoo." But the reality is that "most" problems we face in computer science do not benefit from quantum computers. In these cases, they are no better than a classical computer. But for problems like integer factorization, bringing the compute requirements down to polynomial time isn't just faster: it makes a problem solvable that wasn't before.
Unfortunately [for humanity], there is no evidence yet that quantum computers will solve NP-complete problems efficiently. Most likely, they won't. So don't get your hopes up about solving the traveling salesmen problem any time soon. But there is still a lot of cool stuff we can do with them. In fact, the theory is so far ahead of the technology, that we're anxiously waiting for breakthroughs like this, so we can start plugging problems through known algorithms.