A conventional computer is able to do one thing at a time - that is, read a piece of information (a 'bit') and perform a few basic logic operations on it. These very simple processes form the basis of everything that current computers can do. They do these simple operations sequentially on a piece of information amazingly fast, so it is possible to achieve quite complex outcomes (such as three dimensional graphics and interactive computer games) in almost real time via an unbelievably large number of these simple steps. While the rapid increases in computing power mean these simple steps can be carried out faster and faster, there are fundamental limits to how fast conventional electronic components can operate, and current technology is rapidly approaching these limits. Hence the need for quantum computers.
Quantum computers are different from conventional computers, in that they use electrons and atoms to store information ('qubit') rather than chips. As a result, they can handle billions of pieces of information simultaneously, and so avoid the inevitable bottleneck in conventional computers caused by their limitation to sequentially process one piece of information. Despite the simplicity of the idea, it is actually extremely difficult to build a quantum computer. This is because electrons and atoms can be easily disturbed by their environment, causing the breakdown of information stored in the quantum computer and thus errors in computation.
So, it may be a while before you have a quantum computer on your desk. But physicists around the world are racing to get there, and have already built a 7-qubit-quantum computer, which demonstrates that the physics indeed works.