Just a (not so quick) honorable mention, I think it's important to remember two things:
- This was sort of mentioned before, any of the Eldar who dwelt in Valinor and looked upon the Two Trees, Telperion and Laurelin, were considered "High Elves" in a sense, and always shown with a sort of inner light.
Being one of the first Noldor, Galadriel was born in Valinor (in fact, I would argue she is the oldest living thing in Middle-earth - maybe aside from Tom Bombadil and Fangorn). I'm assuming this is why in the LotR movies, Galadriel glows when the Fellowship firsts meet her in Lothlórien.
- The Phial of Galadriel is not Eärendil's power, per se.
Allow me to explain: Fëanor made the Silmarils in Valinor - most beautiful and powerful of all gems because they captured the light of the Two Trees, and therefore shone with their own inner light. These gems were never able to be recreated.
The reason Galadriel and the other Noldor returned to Middle-earth was to retrieve these gems (as Morgoth, Sauron's master, had stolen them).
Eärendil, the Mariner, ended up in possession of one of the Silmarils, and with it attached to his brow was able to cross the sea in his ship and sail through the enchantments of the Valar, thereby reaching Valinor.
Upon reaching Valinor, the Valar granted his request to send help to Middle-earth, as Morgoth had grown too powerful to defeat.
Additionally, they placed Eärendil (with the Silmaril still on his brow) and his ship Vingilot, into the heavens. Hence, Eärendil (and the Silmaril) effectively becomes the brightest star in the sky.
Indeed, when Morgoth saw this new, brightest star, he feared what was to come. To tie this all back to Galadriel's Phial - the Phial holds water that reflected Eärendil's Star; hence, the Phial holds a reflection of a Silmaril; and the Silmaril is a reflection of the powerful light of the Two Trees Telperion and Laurelin.
Hence, the Phial is a reflection of a reflection of the most powerful "light" ever created in Middle-earth (an embodiment of purity, good, love, hope, etc.) and can be used to combat great evil. I think perhaps the light in Varda's face is the only light more pure (the light of Eru Ilúvatar himself is said to shine from her face).
Alas, as someone pointed out, deep in Mordor the light from the Phial was unable to shine. Here, I assume, such a "diluted" power was not a match for a powerful evil Maia like Sauron. I've always wondered how he would have fared against Galadriel and an actual Silmaril. Alas, they are lost forever and shall only be returned at the end of the world, when they shall be given to Yavanna and broken to release their light and heal the Two Trees.