If you select "auto-attack" on a Fighter it will attack the nearest enemy that is in visible range, and after destruction, go to the next et pp. Ships orbiting on planets will also be attacked, but only, if that planet is also within sensor-range.

After all enemies are destroyed, or none visibly, you'll get this message that auto-attack is off, but actually on the next turn, the ship will go to attack again, if there are enemy Fighters.

Please note that your ship will also engage into hopeless battles, so I would only apply to to superior warships.

In one way, it's a usefull way to clear an enemy system until your Troop Transports arrive, you simply set a fleet in the middle and set it to auto-attack. The downside is that you'll get this popup every turn.