First, begin here: http://www.playonline.com/ff11us/conflict/main/index.html

That is a quick reference to what Ballista (Conflict) is and how to play it. Second, you need to get yourself a license. The requirement to start this is Rank 3. Begin with the 'Ballista License' quest giver in your town. These are the three NPCs:

San d'Oria: Excenmille
Bastok: Invincible Shield
Windurst: Mhabi Molkot

Once you have received the quest from your home NPC, you need to travel to each of the other two realms and see their NPC (shown above). Remember to stop by and see the King/President/Star Sibyl while you're visiting the other towns. They will call you scrawny and say that your home town will not survive Ballista. Nothing new. :)

Finally head home to your starting NPC and receive your license. Next you will want to head to the Ballista Pursuivant in your home town. They will give you all of the information you need to sign up, show up, and fight for your realm!

Ballista Pursuivants:
San D'Oria: North San D'Oria (near Mog)
Windurst: Windurst Waters (near Mog)
Bastok: Bastok Metalworks (1st floor)

When you're ready to fight, go here for a deeper explanation about the game: http://ffxi.allakhazam.com/db/guides.html?guide=219
Also, to watch for your location and times:
http://www.cyberseth.com/ballistamate.html