Short answers, pulled from the story.
The Phial of Galadriel holds water from Galadriel's fountain in Lothlórien that glows with the light of Eärendil's star. This crystal bottle restores Frodo Baggins' senses when all other lights fail and helps Samwise Gamgee fight Shelob.
J. R. R. Tolkien wrote four versions of the chapter Farewell to Lothlórien without mentioning the Phial before adding it in the fifth version. Christopher Tolkien edited these texts later and noted how the object was added late in the writing process after the story had already taken shape.
The light comes ultimately from the Two Trees of Valinor via a Silmaril crafted by Fëanor during the First Age. Galadriel collected the light of Eärendil's Star reflected in her fountain mirror to create this tiny fragment linking Third Age events to tales of the Elder Days.
Jason Fisher draws a parallel between the water in the Phial and the Christian sacrament of baptism while noting similarities between Galadriel and Mary mother of Jesus. Fleming Rutledge comments that the Phial is filled with baptismal water and its divine presence recalls the overpowering light that blinded the Apostle Paul or lit up the face of Moses.
Wētā Workshop constructed physical props for Peter Jackson's film trilogy based on Tolkien's written descriptions appearing in The Fellowship of the Ring and The Return of the King. These props appear when Galadriel presents Frodo with the gift and later when Frodo and Sam use it to get through the Pass of Cirith Ungol against Shelob.