Quantum computing is a field which aims to build a computer based on the principles of quantum mechanics. So, what is quantum mechanics? To explain this, suppose you throw a ball up into the air. The motion of the ball as it rises and falls is well-described by a set of mathematical laws known as Newton's laws, or classical mechanics. Classical mechanics has been around for centuries, its foundations laid by the famous scientist Sir Isaac Newton. Now, what happens if instead of throwing up a ball, you toss something much smaller - say, an electron? Or a photon? Just over a century ago, reknowned physicists such as Max Planck, Albert Einstein, and Erwin Schrödinger were coming to the startling conclusion that classical mechanics fails miserably in describing the behavior of such tiny subatomic particles. To address this, the physics community developed a new set of mathematical laws to describe this miniature world, known as quantum mechanics.
The goal of quantum computing is thus to build a machine which harnesses these new physical laws. How is this achieved? Roughly, in a modern computer chip, a bit is modeled by a transistor, which either has current running through it or not, with each case representing one of the states 0 or 1. In quantum computing, the goal is to replace the transistor, which is a macroscopic or "classical" object, by (say) an electron, which is a subatomic or quantum object. Then, a bit is instead encoded into some property of the electron, such as its spin. To perform a computation, a quantum computer thus manipulates (say) electrons, or quantum systems, instead of transistors, or "classical" systems.