In The Lord of the Rings: The Fellowship of the Ring, what gift does Galadriel give Frodo?

In The Lord of the Rings: The Fellowship of the Ring, Galadriel gives Frodo the Phial of Galadriel.

The small crystal vessel contains the light of Eärendil’s star, captured from the reflection of a Silmaril in the waters of Lothlórien’s fountain. Galadriel gives it to Frodo after the Fellowship leaves Lothlórien, telling him that it will be a light in dark places when all other lights go out.

The phial becomes especially important in Mordor. Frodo uses it to repel Shelob in the pass of Cirith Ungol, where its light overwhelms the giant spider’s darkness. It is not the same object as the Evenstar, Arwen’s jewel, or Sting, Bilbo and Frodo’s elven blade. The gift also connects the trilogy’s events to the older history of Middle-earth, because Eärendil’s light comes from the Silmaril carried across the heavens.