Kreuzberg to Berlin is what SoHo was to London. Before the wall came down, it used to be the most "alternative" area of West Berlin, home to the massive underground and punk movement and large Turkish community.
Throughout the years, Kreuzberg managed to retain everything that's original to Berlin. Those who travel to that city and go to Kurfustendamm (the main shopping street) or Alexanderplatz, miss the real feel of Berlin, that Kreuzberg has.
The coolest underground clubs, quirky cafes, green parks, Turkish kebab shops, lots of people on the streets – all that is Kreuzberg. In summer it's particularly nice, because lots of cafes organize mini open airs and it really feels like everybody lives outside.
The neighbourhood is quite inexpensive too, so a lot of artists and musicians choose Kreuzberg to be their home. However, it's not an expat enclave like Prenzlauerberg (another area in east Berlin) has become. Majority of people are still German and Turkish and there's something really authentic about the area.
Although places such as Schlesisches Tor (where most clubs are) is becoming more "hip", people are moving deeper into Kreuzberg, towards another up-and-coming Turkish area, Neukoelln.
Among areas recommended within Kreuzberg are Goerlitzer Park (and the nearby classic Morena cafe), Oranienstrasse (the central street of the area with lots of cafes, bars, and shops), Schlesisches Tor (most clubs), Graefekietz (a quiet family area with nice restaurants). The famous Bergmannstrasse has become somewhat of a joke, bio everything and home to tourists it's only park of Kreuzberg officially, but in reality it belongs to West Berlin yuppie crowd now.