As everyone else sleeps, Frodo follows Galadriel down to the mirror which can show "things that were, things that are, and things that have not yet come to pass."
Frodo looks into the mirror, which is a silver basin, and sees the destruction of the Shire and the enslavement of the hobbits. Then he sees the Eye of Sauron. It calls to the Ring and Frodo grasps at it, falling backward.
Galadriel says this is what will happen if Frodo fails.
Then she warns him that the "he" will try to take the Ring, and one by one it will destroy the fellowship.
Frodo offers Galadriel the Ring and she becomes frightening in a dark, glowing, deep-voiced sort of way.
She desires the Ring for its power, but the moment passes. She becomes herself again and says she has passed the test and will head west, out of Middle-earth.
Frodo doesn't want to be alone although he knows what he must do.
Galadriel says that to be a ring bearer is to be alone. She is the keeper of the elven ring, Nenya.