New York City became the Big Apple because of a high-class bordello popular in the city in the early 1800s. The founder of the bordello was a Frenchwoman named Evelyn. Locals called her Eve, and jokes about her staff as apples followed. Variations on the apple reference expanded until the entire city was referred to as The Apple by the early 1900s. You can find more details here:

http://salwen.com/apple.html

The origin of "The Big Easy" is less clear, but the leading theory seems to be that it was applied to N'awlins because the city had such a relaxed atmosphere.






(And the perhaps cryptic comment by our friend Faldage is a reference to your use of all caps in your post; it is established internet usage that all caps indicate SHOUTING by the poster.)