While traveling the world map you will occasionally receive a pop-up message detailing a traveler on their way to a tournament or receive an invitation to attend yourself. This quest will let you know where a viable tournament is being held and the duration shown is the time left prior to the tournament being auto-resolved.
There is no penalty for failing to complete this quest, if you learned of it based upon rumor, but if you are invited and decide not to attend you will receive a slight relation penalty from the hosting lord. This applies only if your quest difficulty is set on high.
Once you have completed a tournament you simply need to speak to the hosting lord to complete the quest. If the host is not available within the town then you may turn in the quest with the local arena master.
You can disable the quest by entering the "camp" menu and selecting the "tournament system settings" option. Once in that menu select "Disable Automatic Tournament Quests". You will no longer receive this quest.