Favorite district for Erasmus students and dominated by his Holiness the Berghain, Friedrichshain is truly effervescent. At the very east part of Berlin, the longest piece of the Wall ends up there (East side gallery) and you can still see a lot of buildings which have been through hard times.
When visiting Berlin for the very first time, people use to go in the "East" part. It's a good choice of course, knowing that Berlin is so big that you might need to come back many times. There's actually some very nice areas in the West part, like Shöneberg.