In classical computers, parallel computing is performed by having several processors linked together, so that each processor performs one computation while the other processors are performing other computations.
In a quantum computer, a single quantum processor is able to perform multiple computations on its own by utilizing the fact that the qubit (or quantum bit of information) exists in multiple states simultaneously (a key feature of quantum physics is the ability of the quantum wavefunction to exist in multiple states at the same time). This gives a quantum computer much greater raw computation ability than a traditional computer.
In Deutsch's original interpretation, these extra computations are performed in other universes, allowed under the multiple worlds interpretation of quantum physics. Others, such as MIT's Seth Lloyd, do not feel that multiple universes are necessary (as he describes in his 2006 book Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos).
