# in which you once walked

To say that they "walked in" their trespasses and sins is to say that they often trespassed and sinned. Alternate translation: "because you used to live that way"

# according to the ways of this world

"according to the values of people living in the world" or "following the principles of this present world" 

# according to the ruler of the authorities of the air

"doing what the ruler of the authorities of the air wants them to do." These words expand upon "according to the age of this world." The "ruler of the authorities of the air" is the devil or Satan.

# the spirit that is working in the sons of disobedience

These words expand upon "the ruler of the authorities of the air." This is another way of speaking of the devil or Satan.