Dual clutch "manuals" are automatic transmissions where, rather than a torque converter, two clutches are used to select gears. They are the ultimate evolution of the discrete gear automatic transmission.
Dual clutch transmissions were invented by Porsche in the late 1970s (the year 1975 comes to mind, but I'm not 100% sure), going into race cars in the mid-80s; what is currently on the market as the PDK (
Porsche Doppelkupplungsgetriebe). The idea for dual clutch transmissions, though, came from a fellow named Adolphe Kegresse in 1939, but he never was able to get it to work. Volkswagen brought it to market first, with the 2003 Mark IV Golf R32 and the first in the United States was in the 2003 Audi TT 3.2 V6, under the trade name DSG for "direct shift gearbox".
It was revolutionary at the time; even though DCTs had been used in racing, they hadn't been viable in production cars for decades. And while single-clutch automatic transmissions had also already been on the market, in cars like the Ferrari 355, BMW M3, and Toyota MR-2 Spyder, they were all clunky, user-unfriendly, and basically hot garbage. By using two clutches to preselect the next gear, the DCT made shifting quicker and smoother than that of a single ACT and it was done with less driveline "slush" than a conventional automatic.
While CVTs have struggled with reliability (and that horrible droning noise they make), especially with higher horsepower applications, DCTs are better able to cope with high power now, rendering CVTs the transmission choice for low power economy cars, where fuel mileage is one of the top priorities. DCTs, in my opinion, are the best form of automatic transmission on the market. They do have some issues, however. The clutches are wear parts, for example. And most DCTs are closed units, so it's a hugely expensive job that requires removing the entire transmission to replace the clutches. At least that's how it is on my S4, from what I've read.