LulSec is short for Lulz Security. This group consists of 6 core black hat hacker members, their goal seem to be have fun hacking and exposing the several security threats on the net. They d things apparently for the 'lulz' part of doing it, for the entertainment.
Operation Anti-Security
Operation Antisec, #Antisec or whatever you want to call it is perhaps one of the most mediated hacking operation ever. The two most famous: Anonymous and LulzSec teamed up to bring down and hack targets like Serious Organised Crime Agency (SOCA), Arizona Department of Public Safety and newspapers like The Post and The Times.
LulzSec and the FBI
After hacking Fox.com and leaking X-Factors contestants personal info, they hacked onto The American PBS system and even Sonny Pictures was not sparred. Now they are going big - FBI Affiliated Organisations. You can get more info on all their attacks on Wikipedia. I came across a video relating some interesting info about the group and FBI, you are welcome to take a look.
I hope you have as much fun as i did when finding more about the group.
Their Tweeter page: :http://twitter.com/#!/LulzSec
Download all their leaked/published documents: http://thepiratebay.org/user/LulzSec/
Their Last Release: http://pastebin.com/