This manual page explains the turn mechanics in PW.
A faction can launch an invasion if it has the initiative and there are no Challenges to respond to. Once these conditions are met a timer will begin, the faction has that long to decide upon which planet to invade and gather the commanders required to issue a Challenge.
The initiative timeout period begins at 30 minutes but can extend to up to 90 minutes if the faction loses many invasions due to inactivity in a row while failing to issue and Challenges. This is to prevent a faction from losing many invasions during periods of inactivity.
The turn on the galactic map is increased whenever an invasion occurs. That is either a Challenge has been accepted or has won automatically due to a lack of defenders. This causes all 'per turn' effects to occur such as dropship production and Influence spread.
Each faction receives 100 metal per turn. This metal is given to the players involved in the invasion. If no players are involved in the defense of an invasion it is given to the faction as a whole so can only be used by economy managers.