The princess has been captured - the empire is trying to do a mission to turn her to the dark side - the rebellion is doing a mission to save her. Which mission is completed first? If the rebels succeed in their attempt to save her and the empire succeeds in turning her- what happens?

Fortunately, the missions are not played concurrently. During the Command phase, each side takes a turn in attempting a mission or activating a system. The Rebels get the first turn in the command phase, so they get to try and rescue her first. The Empire can always send a leader to oppose, if that leader is ready and waiting in the leader pool. Should the rebels succeed, Leia and her rescuers flee to the Rebel Base space and the Empire will just have to pass on the mission to turn her to the dark side.

Right, who ever plays their mission first gets the first attempt.

The sequence would be

1. empire captures the princess
2. rebels get a turn and may attempt the rescue mission
3. if the rescue mission fails then on the next empire turn they may attempt to turn her to the dark side.