Eos and Dione are the 2 factions of the game. You pick one and you will fight alongside fellow members of that faction and against the other faction in Realm vs Realm war. You can join guilds that are only of that same faction and you can switch sides at the cost of 50k gold and losing any battle points you may have acquired.
PK = Player Kill. Attacking another player. You can activate PvP with the double swords under your mana bar and attack anyone else that is in PK mode.
Daily quests are scattered throughout the maps with 5 daily quests for Ledise and Raden. Daily quests reset at midnight so they can be done once per day only. That's a total of 10 daily quests per day if you are high enough level between the first 2 major cities.