What are autothrust and manual thrust on an A320 (or other aircraft)?
Manual thrust means that the engine thrust corresponds directly to the position of the thrust lever.

Autothrust means that a target speed is dialed into the system (using a knob that is usually located on the glare-shield) and the auto-thrust system automatically adjusts the engines to maintain that speed.

Now in an aircraft with classical controls the longitudinal stability means the aircraft tends to maintain speed with pitch and pilots basically need to use thrust to climb or descend and yoke to adjust speed, always making adjustments with the other control as well to avoid oscillations and compensate for the more complex parts of the power-pitch relationship.

Due to this, it is usually recommended to fly either fully manual or fully automatic (autopilot+autothrust).

But Airbus A320 and newer (A319 is just a variant of A320) changes that with the artificial neutral stability in normal control law. Even with autopilot off, the flight computer automatically adjusts trim so that the aircraft flies straight so that adding power makes the aircraft accelerate and reducing it makes it decelerate. However pushing the stick forward to descend will still make the aircraft accelerate and pulling it to climb will make it slow down.

Due to this, Aibrbus aircraft are easier to fly with autothrust always on (except for take-off roll; autothrust is disabled when the power levers are advanced above certain position, which is only done for take-off and in emergency where extra power is needed).