These two dragons are Timed Events that occur daily. The event works by a large number of players having their cards with the highest damage scores all attack the dragon. The event will automatically end after an hour or once players have dealt enough combined damage to kill the dragon. The cards each get one hit and one attack with their Nirvana (or special attack) then the dragon attacks and usually kills all cards. After the attack is complete you will be shown how much damage you managed to inflict on the dragon and you will be rewarded with half your damage score in Coins each turn.
Acnologia during battle with Auto battle turned on
Boss Dragon when event is turned off
Boss Dragon begins at 11:00 and finishes at 12:00 (GMT+8), while Acnologia begins at 20:00 and finishes at 21:00 (GMT+8). (*Although these are the given finish times, the dragons are usually killed before this time is up.) Each dragon in every server begins at level 5, but if either dragon is killed sooner than half an hour during the battle, that dragon in it's respective server will level up and become more difficult to kill.
The idea behind it is to deal the most damage to the dragon so you receive more coins and gain a high rank among other players for even more rewards. Once your cards have been killed there is a one minute cool down period before you can attack again. The cool down period can be over-ridden, however, by spending 20 Diamonds to resurrect your team and attack immediately. This will increase the overall damage you manage to inflict on the dragon and therefore offer a chance at increasing your ranking against other players.
If you pay to reach VIP4 or higher there is an "Auto Battle Function" which, when selected, will automatically begin a battle with the Dragon when your cool down runs out as long as you leave the Dragon battle running.
Certain cards, such as Gildarts have perks which help to improve your score. These cards are useful to keep, even if you do not usually use them, just to play them during Dragon battles.
There is an additional reward given to the player that deals the killing hit "slayer" to the dragon. This killing hit can be achieved with good timing and luck.
For the both Dragon the Gold Coin ranked rewards are:
1st: 500,000 coins
2nd: 400,000 coins
3rd: 300,000 coins
4th - 10th: 200,000 coins
Slayer Reward: 300,000 coins
BOSS Dragon additional rewards are 5s Jigsaw Bags .
Acnologia additional rewards are 4s Equipment.
*Please note, there is a typing error within the game were the Boss Dragon is listed as "Off Dragon".