A LAN party is a party for nerds. Everybody brings their computer and they all get hooked up into a local area network, or LAN. It's basically like their own little version of the internet; computers in a LAN can share files, play games with each other, etc. Typically you play games with everybody present at the LAN party for one day at the very least. I just had a small one in my garage a few weeks ago, 8 people present. It was a blast. We spent all of saturday playing Starcraft, Battlefield Vietnam, and Wolfenstein: Enemy Territory; had a DDR tournament in the middle of the night on my PS2 (in which I ranked third); and then we played Battlefield Vietnam until about noon on Sunday. It was a BLAST.
LAN parties take a lot of preparation and time, but they are very much worth it. You should go to one.