Operation of a quadcopter/multicopter is similar to that of a helicopter. It has a control similar to a helicopter's cyclic control that enables pitching the quadcopter anywhere within its 360o radius. But the quadcopter differs from the helicopter in that it has no tail boom. Because of this, the quadcopter can easily fly in any direction regardless of which way its "front" may be pointing. Thus a quadcopter can change its direction of flight more rapidly than any other flying machine.
The multirotor concept has been around for a long time. But until recently it wasn't practical as the design is inherently unstable. Controlling multiple rotors to maintain steady flight is extremely difficult if not impossible for the average person. But advances in gyroscope attitude sensing technology (thanks to the Wii, tablets, and phones) have developed very small, low-power attitude sensors that can be incorporated with computer logic to automatically balance the quadcopter in flight. Thus the pilot can now focus entirely on his flying skills, and no longer be distracted to also manually balancing the quadcopter from flipping over.
Currently there are two versions of attitude gyros that you should be familiar with. They are titled as "3 axis", and the somewhat incorrectly titled "6 axis". The 3 axis automatically balance the forward & aft tilt (pitch), and right & left tilt (roll) of the aircraft. The right and left turning axis (yaw) is also steadied by the sensors. A balanced and properly trimmed (more on trimming later) quadcopter should takeoff and hover relatively motionless in the air. It should not tilt or turn in any direction until you apply control to the aircraft. Additionally, if you apply and remove control, it should slowly but automatically return to a horizontal non-turning position.
It's obvious that for most beginners a 6 axis quadcopter is the best option as it's most forgiving. It's even appreciated by more advanced flyers as a 6 axis quadcopter is also very nimble, able to "turn on a dime".