What is the name of the gearbox that combines manual and automatic features with two clutches?

The gearbox that combines manual-transmission principles with automatic operation and two clutches is a dual-clutch transmission (DCT).

A DCT is built around two independently controlled clutch assemblies. One normally handles odd-numbered gears and the other even-numbered gears. While one gear is transmitting power, the next gear can be selected on the alternate shaft; opening one clutch while closing the other produces a rapid shift. The driver may let the electronics choose gears automatically or select them manually through paddles or a lever.

Because it uses gear pairs and clutches rather than a conventional hydraulic torque converter, a DCT is mechanically related to two automated manual gearboxes sharing one housing. That explains both its quick shifts and its manual-like construction. The trade-offs can include low-speed hesitation, clutch wear, added complexity, and less smooth behavior in traffic than some traditional automatics.

DCT is the broad technical name. Manufacturers use brand names such as Volkswagen DSG, Audi S tronic, and Porsche PDK for related implementations.