There are quite a few areas and places where you can party.
Altstadt is mainly for tourists. If you want to get drunk fast with fellow travellers you are most likely to meet them there. Peters Brauhaus provides good food and not so good beer, the Paeffgen Brauhaus vice versa.
Kwartier Lateng (Zulpicher Straße) is for students and young persons who want to drink and have a good time. Party and concerts also in the Prime Club/Luxor, the Blue Shell and the mtc.
Friesenviertel is for the more well-to-do people. Make sure to check out the Paeff (not to confuse with the Paeffgen!). Often there are quite good DJs at work.
Ehrenfeld is for students and more alternative folk. Lots of nice pubs, bars, clubs and more. The Underground, the Live Music Hall (rock, alternative), the Sensor Club (dance), the Art Theater (funk, Drum'n'Bass), the Sonic Ballroom (punk, indie) are must-sees when you are in Ehrenfeld.

Besides that there are a few cool clubs and classic bars. Schreckenskammer and Elektra in Altstadt-Nord, Gebäude 9 in Deutz, Westbahnhof, Tor 2 right next to the central station, Apropo in the Severinsviertel,.....