The idea of Quantum Computing was first proposed by Feynman based on the idea that simulating a quantum system would be a hard problem for a classical computer. The logic goes somewhat like this. Consider N quantum particles. At the next time step there are approximately N possibilities. After N time steps the number of possibilities has gone up to N^N. Thus simply storing all possible alternatives requires memory which goes up exponentially.
On the other hand a quantum system could easily simulate another quantum system. This idea was put in a more concrete form by Peter Shor in 1995 with Shor's algorithm which showed that it would be possible for a quantum computer to factorize a large number using a polynomial time algorithm.
As the fact that factorizing a large number is a "hard problem" for a classical computer is the basis of RSAencryption, this discovery spurned off a lot of research on Quantum Computing.
A lot of research is being done both in trying to realize a quantum computer using technologies such as NMR and in developing new algorithms. More information and links may be found at http://www.qubit.org.