Alberic, your mentor, is a former Azure Dragoon, and he initially asked for help from Gridania regarding a delicate matter. Since secrecy was an issue, the Lancers' Guild chose to send you, an adventurer--someone who's both skilled and has no political affiliation. He briefs you on how a priceless relic has been stolen and he needs your help in tracking down and apprehending the fugitive.
The fugitive is Estinien. Estinien's shocked to see that The Eye is reacting to you, and departs after warning that you'll meet again.
Alberic is equally shocked to hear that The Eye reacted to you, and decides you deserve to hear more about what you're up against. You're told that Estinien is the Azure Dragoon, chosen by The Eye. Since you've also been chosen by The Eye, Alberic decides to train you so that you'll be able to hold your ground next time you encounter Estinien, and reclaim the relic.
Eventually, Estinien arranges a meeting with you and Alberic. Alberic asks him why did he steal the eye from the Holy See, knowing that it would doom Ishgard by awakening Nidhogg? Estinien corrects him saying the prevailing beliefs concerning the eye were wrong and he sensed Nidhogg was already rousing, so he stole the eye to save Ishgard in hopes of using it to lure Nidhogg away--an act of self-sacrifice. But now that you're in the picture, another potential Azure Dragoon where never in history has there been more than one, Estinien proposes that together you may be able to slay Nidhogg instead. He leaves your preparation to Alberic.
Alberic sends you on life-or-death trials to retrieve and bond with drachen armor, as all Azure Dragoons need to do. The armor is forged using the blood of dragons, and as blood calls to blood, this armor is needed to awaken your "inner dragon". (Note, they don't say anything like you need dragon blood in your veins, they merely say that the dragon blood would "resonate" with yours.) He can't just give you his old armor because his is already bonded to him.
Before you could get the last piece of armor, Estinien sends word that Nidhogg is already moving, so if you're to fight, it's now or never. Alberic hurries with you because he's desperate to tell Estinien the whole truth of his own encounter with Nidhogg, or else he may fall prey to the dragon's will. People know that in the past when Alberic was the Azure Dragoon, he warred with Nidhogg and fought him for three days and three nights. Exhausted, Alberic wagered everything on a single blow and stabbed his lance into Nidhogg's empty socket. Nidhogg screamed in agony and flung Alberic away, retreating in defeat while also leaving Alberic injured and unfit to continue as an Azure Dragoon. What people don't know is that while Alberic and Nidhogg were locked together in that final blow, their eyes met and Alberic was overwhelmed by Nidhogg's emotions.
"Sorrow... Rancor... Pity... They threatened to drown the man in me and leave behind a dragon."
Alberic voluntarily gave up being an Azure Dragoon to retain his humanity, rejecting the power of The Eye. As Nidhogg fled, the remainder of his horde laid waste to Ferndale, and Alberic, no longer a dragoon, no longer had the ability to save it. Estinien was the sole survivor and Alberic chose to raise him as his own, out of guilt, because just as Nidhogg fled, he had fled from being a dragoon, and Ferndale paid the price for it.
When you catch up to Estinien, he's blinded by grief and rage and wants Alberic dead. The Eye had already revealed the past to Estinien--of Alberic's failure to protect Ferndale--so Alberic doesn't get a chance to warn him about the risk of being enthralled by Nidhogg.
You defeat Estinien, but maybe already lost to Nidhogg's will, he looks to the sky and beseeches Nidhogg for strength. Estinien is consumed by a dark aura. Alberic flinches at the blow that was meant to kill him and opens his eyes amazed to see that he's alive, protected by the shining visage of the first (and strongest) Azure Dragoon, Haldrath. Haldrath strikes down Estinien with a Dragonfire Dive. Afterwards, you're there in Haldrath's place, and Estinien is nowhere to be seen.
In the aftermath, Alberic tells you that you awakened to the ability that only Haldrath possessed and summoned Haldrath's drachen mail (now yours) which was apparently sealed within the soulstone. He posits that your indomitable sense of justice must've resonated with Haldrath's.
In the meantime, he prays for Estinien's safety. Whatever happened in the past, he loves Estinien as his son. And when Nidhogg's shadowy aura brushed against Alberic...