Audi quattro (they insist on small letters) is a four-wheel drive technology that was created for rally conditions, nowadays applied in their flagship products with great success, most notably the TT quattro and S quattro cars.
Whereas most on-road cars are only driven by either their front wheels (almost all smallish cars) or their rear wheels (all "sporty" cars, bigger cars, e.g. most BMWs and Mercedes-Benz cars), Audi's premium cars have a drive system that takes the engine power to all four wheels, contributing towards better stability and traction on the road.
The first ever Audi quattro car took the world rally scene by storm in the 80s and is today considered to be one of crown jewels in Audi's Vorsprung Durch Technik philosophy.
At the heart of quattro ia s "torsen" (torque-sensing) intelligent differential that splits torque between the front and rear wheels based on the driving conditon. For example, if your front wheels were over snow, then quattro would send more power to the rear wheels to elliminate slip and improve traction.
It is also interesting to know that Audi's philosophy says that rear wheel drive is inherently unsafe and they only manufacture front-wheel drive and quattro cars. Audi also has a unique, enhanced front-wheel drive setup called FronTrack.