DCT: Dual clutch gearbox is called DCT for short, because it has two sets of clutches, so some people call it "power shift".
The difference between DSG and DCT: the electronic module of gear shifting is different. The 6-speed one is from Borgwarner, USA. Borgwarner is called DCT gearbox. Later, because of the need of publicity, all automakers were called DCT gearboxes.
DSG is the name of Volkswagen in Germany, because Volkswagen also has a 7-speed gearbox. In Volkswagen, the 7-speed DSG is called DQ200, while the 6-speed DSG is called DQ250, which is the 6-speed DCT of Borg Warner.
To sum up, DSG generally refers to Volkswagen's 7 speeds, while DCT generally refers to all 6 speeds.