Darth Vador captured the princess and Lucas and his group came to save her. Though Darth Vador isn't after the princess, however, he is after his old master. Darth Vador wants to see his old master face to face and have a talk with him since he know his old master isn't planning to escape. Death star is a powerful weapon that can almost kill anyone.

Luke Skywalker joined the rebel forces and helped them fight the black forces. Darth Vador was after the princess because he wants to the plan. The plan shows how to work the death star. But then the plans were actually in the robot.