So SPOILER ALERT.
This is a really big event that kicks of the expansion and sets the dire tone for what follows. Ultimately, the last two expansions were sort of lower-stakes than the previous ones - failure would mean conquest, not annihilation. This time, though, things are the highest stakes we've ever had, and big things happen to establish that fact.
We arrive at the Broken Shore in an attempt to head off the Legion invasion. Much of the Alliance leadership is already there, and we've gone to reinforce them.
We find Jaina and Genn Greymane (who has an awesome new model in the cutscenes that I really wish player Worgen could get - an expressive mouth and everything!) fighting the demons on the beach. With our help, they rendezvous with Mekkatorque (in a sweet mech-suit) and Varian, who are pushing farther up.
We fight our way through a growing city of demonic buildings until we discover that Tirion Fordring and his Argent Crusade forces have been captured. Gul'dan tortures Tirion and then has a massive doom lord (who I believe is a boss in the Nighthold) blast the Highlord with Fel Magic until he appears to die (the only reason I know he hasn't is because of the Retribution artifact quest.)
The Alliance and Horde trade off fighting Krosus while the other deals with adds. When the doom lord is defeated, we go up a rise to confront Gul'dan.
But Gul'dan manages to summon a massive army of demons - including many demons that we've already "killed" like Jaraxxus, Brutallus, and even Tichondrius (Hakkar the Houndmaster's even there, who we've heard about for a long time but never seen in game. No relation to Hakkar the Soulflayer.)
We fight them off, but in the midst of battle, Varian calls on Sylvanas' archers to provide covering fire. He looks up, only to see the Horde forces receding from the cliffs in an apparent betrayal at the worst possible time.
Mekkatorque calls in a gunship so that the Alliance forces can retreat just as an incredibly massive Fel Reaver begins to attack. Genn climbs aboard while Varian hangs from a rope ladder. But the Fel Reaver grabs hold of the ship, threatening to tear it apart as soldiers roll off its deck to their deaths.
Varian takes out the letter he had been writing to Anduin and hands it to Genn. He then dives off the ladder and drives his sword into the Fel Reaver, ultimately destroying it.
As the gunship flies away, Varian makes his final stand, slaying demons left and right, but ultimately, two felguards jam their spears into his back. As Varian falls to his knees, mortally wounded, Gul'dan approaches him and asks him "How does it feel to die for nothing."
Varian replies "For the Alliance."
At that, Gul'dan channels his fel magic into Varian, burning him from the inside until his body explodes in a burst of green fire.
You know, I had my problems with Varian when he first showed up, but damn if that ain't a heroic sacrifice. I'm honestly kind of sad now. I only hope his soul was able to survive the warlock's magic so we might see ghost Varian at some point.
Following this disastrous defeat, Genn tells you that he can't stand to face the young King Anduin, and so sends you with Lorna Crowley and his daughter Tess to deliver the letter to the new king.
At this point, the story syncs up with that of the Demon Hunters. You deliver the letter and after Jaina leaves - unwilling to focus on the Legion and let the Horde be - an Illidari arrives to warn you about the Wrathguard assassins in the keep.