After entering Ustengrav and discovering that the Horn of Jurgen Windcaller is missing, you follow the instructions on the note and meet Delphine in the Sleeping Giant Inn in Riverwood. She needs your help  stopping whatever is bringing all those dragons back to life. The next place where a dragon will be resurrected from its burial place is Kynesgrove, according to her research. Before heading out to Kynesgrove with her, it is recommended you go to High Hrothgar to complete the previous quest, "The Horn of Jurgen Windcaller". Once you've sorted out the business there, you can proceed with  the next quest in the main story campaign, "A Blade in the Dark".

Open your quest journal:
A Blade in the Dark
Delphine believes that dragons are coming back to life and that the next one will resurrect near the town of Kynesgrove. Once I've convinced her that I'm Dragonborn by killing the dragon, she promises to fully explain who she is and what she wants with me

Locate the dragon burial site
Talk to Delphine

Kynegorve is south of Windhelm. Either fast travel to Windhelm if you've been there before, or hire a carriage from Solitude or Whiterun. From Windhelm, take the road south to Kynesgrove. Then take the path east, from behind the inn. On your way you will see the Dragon Alduin! Reach Delphine on top of the road. She will tell you to take cover since a dragon is attacking. Alduin will resurrect a dragon, Sahloknir. Your goal now is to kill the dragon Sahloknir. Defeat the dragon and absorb its soul.

"Near Kynesgove, we saw a dragon shouting another one back to life. That dragon is dead, and now it's time for Delphine to give me some answers"

Speak to Delphine. Ask her who she is and what she wants with you. She will tell you she is one of the last members of the Blades, dragonslayers who used to serve Dragonborns. For the last 200 years, since the last dragonborn emperor, the Blades have been searching for a purpose. Now that the dragons are coming back, their purpose is clear again. They need to stop them. Ask her what she knows about the dragons coming back. She will say she was just as surprised as you to find the black dragon here. Tell her you've seen that dragon before, the one that got away. That it was the one that attacked Helgen when Ulfric escaped from the Imperials.

Ask her who the Blades are. She will be upset noone remembers their name. For the last 200 years they've been searching for a dragonborn to guide and guard, but never found one, until now.

Ask her what's our next move. She will say  the first thing they need to do is figure out who is behind the dragons. The Thalmor is a good bet. She believes they are involved. Ask her who the Thalmor are. She will tell you they are the faction that rules Aldmeri Dominion. The ones who almost destroyed the empire during the great war, thirty years back. There is no worse enemy to human kind in Tamriel. Ask her what makes her think the Thalmor are bringing Dragons back. She will say she doesn't have solid proof, but has a gut feeling. Since the empire captured Ulfric, the war was over. But they wanted to stir the war back on, so they used the dragon to free Ulfric and reignite the war. And now the dragons are attacking everywhere, indiscriminately. Skyrim is weakened, the empire is weakened. And the Thalmor stands to gain from that.

Ask her why the Thalmor are after her. She will say that before the great war, the Blades helped the empire against the Thalmor. Tell her you need to find out what the Thalmor knows about the dragons and if she has any ideas. She will suggest getting into the Thalmor embassy, their center of operations in Skyrim. But that place is heavily guarded. Ask her how you could get in. She will say she has a few ideas, but she is not sure yet, and to meet her in Riverwood. She gives you a key, Delphine's Secret Door Key which you can use to get into the secret hideout in the Sleeping Giant inn.

This completes the quest "A Blade in the Dark" and starts the next quest in the main campaign, "Diplomatic Immunity"