The Fellowship of the Ring was founded at the Council of Elrond with the purpose to aid Frodo in his attempt to destroy the OneRing:
also called the NineWalkers, The Company of the Ring.
The first half of FotR reveals a bumbling and impromptue "fellowship" of both Frodo Baggins and several of his cousins and gardeners who conspired against him and Bilbo to discover why they seemed curiously unavailable when unwanted company dropped by. Hobbit stealth when invisible matched Hobbit stealth when hunting.
See also: A Long Expected Party
The second half of LotR revealed Elrond's ability as a leader. When Frodo takes on the Quest to destroy the OneRing of Power, Elrond arranges his escort, but lays only on Frodo the bond that he must bear the ring directly to MountDoom and toss it in. The rest of the team are under orders to follow the chance meetings on the road; to leave Frodo if that supports him best. Of course Elrond recognizes the forces that could fission a team enduring open warfare, even without an evil ring in their midst, subconsciously advertizing itself to them as a great weapon.
The savage irony remains Elrond's greatest act of leadership, forming the Fellowship, was his last. His own Ring's ability to withstand change, though untainted by the "hand" of Sauron, still depended on the permanence of Sauron's own Ring.
See also: Bookshelved:FellowshipOfTheRing for a literary deconstruction.