On this page of our guide to TES V: Skyrim we have prepared a detailed description of (The Blades) Ask Esbern about dragon lairs side quest. Our walkthrough explains how you can unlock this quest. Also, you will get to know how to reach randomly chosen dragon lairs.
You can approach this quest only after finish the mission of brining three volunteers to the Blades, as part of (The Blades) Bring a follower to Delphine. In return, Delphine should send you to Esbern. Find the old man and ask him of his study on dragon lairs (screen above). Esbern will inform you that he indeed did discover a lair and will send you to slay the dragon. You will also learn that those who you have recruited will help you in this mission.
Notice - Before leaving Sky Haven Temple, it's worth to ask Esbern of the free Dragonslayer's Blessing. It stays active for five days and increases you chance of critical hit while fighting dragons by 10%. You can of course return to Esbern and ask him for another blessing after the effect wears out.
Open the world map and head to the marked dragon lair in a randomly chosen area of Skyrim. The whole fight with the Dragon (screen above) will be no different than all the previous ones, so base on well-known tactics. The battle will be of course made easier thanks to the allies who you can count on. Afterwards I'd recommend examining the corpse of the dragon (the Dragon Bone and Scale will come in handy in the next mission) and return to Esbern afterwards. The old man will have another mission for you - (The Blades) Bring a Dragon Scale and a Dragon Bone to Esbern.