Can autopilot navigate automatically?
Generally speaking (not specific to Airbus), what the layman refers to as "autopilot" is actually three logical systems:


The Flight Management System (FMS), which  figures out where the plane is in relation to the supplied flight plan.
The Flight Director (FD), which figures out how to navigate the plane to where the FMS wants it or according to manual inputs.
The Autopilot (AP), which manipulates the flight controls to follow the FD.


Generally, the FMS and FD will be engaged for the entire flight, but the AP is not engaged until after takeoff and is disengaged shortly before landing.

While the flight plan is issued before takeoff, it is frequently amended by ATC during flight. When this happens, the pilot just enters the new data into the FMS and lets the FD and AP adjust as needed.

However, sometimes ATC will issue "vectors" to aircraft, which is an instruction to fly a particular heading or track. To comply, the pilot dials that setting into the FD control knob and changes modes so it follows that manual input instead of the FMS. When ATC puts them back on course, the pilots simply switch the FD back to following the FMS.