Elrond has called a council of important men, elves, and dwarves.
Boromir is there and tries to touch the ring but Gandalf stops him.
Boromir wants to use it against the enemy, but Strider tells him that is impossible: the Ring only has one master.
We officially learn that Strider is, in fact, Aragorn: heir to the throne of Gondor. With Boromir being the son of the steward, we sense a little bit of tension.
Then Gimli, a dwarf, tries to break the ring with his axe, but it's his axe that breaks. The only way the Ring can be destroyed is by casting it back into the fires of Mount Doom in which it was forged.
Someone needs to take the Ring to Mordor but, as Boromir famously claims, "one does not simply walk into Mordor."
The company begins to argue amongst themselves about who should take the Ring (dwarves and elves don't really get along). During the argument we see Frodo, who is greatly affected by the Ring. It's almost like it's calling to him.
Finally Frodo offers himself as the Ring bearer and everyone is silenced.
Aragorn, Gandalf, Gimli, Boromir, and an elf named Legolas pledge their allegiance to Frodo on his quest.
Also, Sam, Merry, and Pippin show up and say they're going too.
So it's settled. This company of nine will be the fellowship of the Ring.