The Dragon quest is started at any Dragon Quest Director at the noob starting areas.
The quest involves 10 steps. These are usually 1 of 3 possible options.
1 - Talk to a NPC.
2 - Give NPC 10 DQ items.
3 - Kill 10 mobs.
At the end, you have to kill a miniboss mob, usually not too tough.

The Reward?
You get a Dragon order (Bronze, Silver or Gold) depending on the level of the DQ you do. The Dragon Orders are collected and used to purchase "rare" mounts.
NOTE: The "rare" mounts can now be bought from the cash shop or from other players ingame. Just selling the DQ items actually nets you more coin than if you tried using them for Dragon Orders.
HOWEVER: Doing Dragon Quests also gives you XP. Do I think it's worth it? No.