Here comes the science
What is a quantum computer?
A quantum computer would be a computer that works at the sub-atomic level and is based upon the bizarre laws of physics that occur in the minute world of quantum mechanics. Parts of a quantum computer would be so small some could only be seen under a microscope.
Why do we need them?
Need them or not, they are likely coming anyway due to physical world limitations on Moore's famous Law, which has driven the development of computing as we know it and explains why computers and other digital devices based on microprocessors keep getting cheaper, faster, more powerful and more compact. Postulated by Intel cofounder Gordon Moore in 1965, Moore's Law states that the number of transistors that can fit on a microprocessor will double about every 18 months to two years, with computing power also increasing accordingly. But eventually transistors cannot get smaller, nor communicate with each other, using materials and concepts at the levels of the world we interact with - the classical world of Newtonian physics. They will have to move to the level of atomic and sub-atomic particles - at which point, the wacky laws of quantum physics kick in.
Also, a computer based on quantum laws would be exponentially more powerful and versatile and could perform calculations at a speed we can barely imagine.
How can they do this?
Because in the quantum world, matter can be in any number of states simultaneously. Theoretically, a quantum computer could perform a huge range of calculations simultaneously.