Invented by a Frenchman just before WW2 and introduced to road cars in 2003 in the Golf R32, this type of transmission is a semi-automatic gearbox - two clutches take care of two sets of gears. One clutch takes care of odd gears (1, 3, 5, 7) and the other clutch deals with the even gears (R, 2, 4, 6).
How it works:
One clutch has 1st gear engaged, with the car moving forwards. Meanwhile, the other clutch has 2nd gear ready to be engaged. Come change-up time, clutch no.1 disengages 1st and clutch no.2 engages 2nd gear and the car keeps going, whilst clutch no.1 prepares 3rd gear for engagement. All of this happens within milliseconds, so power delivery to the road is uninterrupted.
Usually - though not always - dual-clutch gearboxes are more fuel-efficient than an equivalent manual gearbox.
For more info, see Wikipedia.
"What gearbox have you got in your Audi TT?"
"Oh, I've got Audi's (VW's) dual-clutch gearbox. It's fantastic!"