As a previous poster said, Carnival has the "party atmosphere" and tends to cater to younger travelers. Royal Carribean is a step up and still has many young people.
Princess tends to be good value for your money, but the clientele is generally older. If you are in your 20's, you might be bored.
I have never sailed on NCL, but I hear that the clientele is generally older and nightlife stops after 10 pm.
Would also recommend you check out
www.cruisecritic.com.
Tips - Do not buy your tours from the cruise company. They are VERY overpriced and you will have many options once you land - especially in the Carribean. Consider hiring a taxi for a day. We did this in Jamacia for $80 and got to see all the sights the tour companies did, plus some they did not...
Don't forget to budget for "extras" on your cruise. Drinks (even non-alcoholic beverages like flavored coffees) are quite costly and add up quickly. There is also the casion, bingo, horse races, etc. $20 here, $50 there - your cruise bill could easily be more than your trip.