What is the difference between manual, automatic, DSG and CVT gearboxes?

They all match engine speed to road speed. The differences are in how they change ratio, and each has a distinct character.

Manual. You select gears with a lever and disengage drive with a clutch pedal. Mechanically simple, cheap to build and repair, light, and efficient. The clutch is a wear item requiring eventual replacement. Fully under your control, which some drivers prefer and which matters for engine braking and towing.

Traditional automatic (torque converter). Uses a fluid coupling — the torque converter — instead of a clutch, with a planetary gearset providing ratios. Extremely smooth from rest, and the fluid coupling multiplies torque at low speed, which is why these suit large engines, towing and heavy vehicles. Historically less efficient because of slip in the converter, though modern lock-up clutches and eight-to-ten-speed designs have largely closed that gap.

DSG / dual-clutch (DCT). Essentially two manual gearboxes in one housing, each with its own clutch — one handling odd gears, one even. The next gear is pre-selected while the current one is engaged, so shifts happen by swapping clutches in milliseconds. Very fast and efficient, which is why performance cars use them. The weakness is low-speed behaviour: creeping in traffic, hill starts and parking manoeuvres can feel hesitant or jerky, because there is no fluid coupling. Dry-clutch versions are more prone to this and to clutch wear.

CVT (continuously variable). Uses a belt or chain running between two variable-diameter pulleys, providing an infinite range of ratios rather than fixed steps. This lets the engine sit at its most efficient speed regardless of road speed, which is why it is common in hybrids and economy cars. The characteristic complaint is the rubber-band effect — engine revs rise and hold while speed catches up, which many find unpleasant. Manufacturers often program artificial "steps" to mask it.