An idealized classical computer deals with arrays of bits. A single bit is an object with two states. An array of states just means we're working with an ordered list of bits. If you have n bits of storage, then your system can be in any of 2^n states.
In a quantum computer, things are a bit different. Individual qubits don't have a definite value, only an amplitude for eahc possible value. Even more intriguing is that qubits can be entangled. This means that we don't just assign each qubit its own amplitude, but we assign an amplitude to each possible state of the entire array. Measuring the array collapses the quantum state to a classical state.
On the surface, it seems like it might not benefit you. With an n-qubit array, you can store 2^n states, but as soon as you measure it, your state is lost. But there are little tricks you can use to speed things up. Functions which act on quantum registers are essentially able to apply the function to every possible argument simultaneously. For certain problems, if you set up the right entangled state, run the right function, and then "undo" the entangling, you can obtain the answer in fewer steps than any classical algorithm.
The simplest such algorithm is called the Deutsch-Jozsa algorithm.