It's been many years since I took an introductory class in quantum computing, but basically the way computation works with quantum computers is fundamentally different. I would require me relearning a lot to explain how it makes computation faster, but basically it's possible to devise algorithms where multiple solutions are returned as a superposition of states, which when observed will usually collapse to the most probable answer, which is the correct one.

The algorithm I studied was for prime factorisation. Basically it lets you calculate prime factors of a number in polynomial time which ultimately breaks the security of things that depend on prime factorisation being hard (slow), namely all public key encryption.

Shor's Algorithm: http://en.wikipedia.org/wiki/Shors_algorithm