Many Meetings is the first chapter of the second book in The Fellowship of the Ring.
When Frodo awakes he is in the house of Elrond at Rivendell. Gandalf is there. Elrond has called a council meeting to discuss what must happen. Gandalf tells Frodo that Strider is also called Aragorn, Man of the West, and that the Black Riders are also known as Ringwraiths or the nine servants of the Lord of the Rings.
Frodo finds that for the past four days and three nights he has been delirious. He learns that Elrond, a master healer, has been at work trying to save him. The rider that wounded Frodo did so with a Morgul-knife, and a piece of the knife was still stuck in Frodo's body. That piece had worked its way toward his heart, but Elrond had removed it. If the fragment had reached Frodo's heart he would have become like the Riders, a servant to Sauron. The Ring would have been taken from him, which would have been a disaster for Frodo.
Frodo learns about all those who work for the Dark Lord. The men are made of nothing, and wear robes to give shapes to their otherwise shapeless existences. The horses, just like many Orcs, trolls, kings and men, have been brought up under the Dark Lord's power and they are in his service forever. Gandalf tells Frodo that the white figure he saw at the river was Glorfindel and that the river itself was commanded by Elrond.
After they have eaten, they go to the hall of fire. To his delight, Frodo discovers Bilbo is living there among the others. Bilbo tells Frodo what he has been up to and asks to see the Ring. When Frodo produces it, Bilbo at once asks him to put it away, fearful of the way he has responded to it. Strider returns and Frodo discovers that he is also called Dinadan.