Galadriel was the last of the Noldor in Middle-Earth, who came to Beleriand in the First Age (lead by Fingolfin) to fight Morgoth. When Morgoth was defeated, most of the Noldor were forgiven, but some Noldor still chose to stay in Middle-Earth instead of returning to Valinor. Galadriel was one of them, Gil-Galad and Cirdan were two other examples. (although Cirdan wasn't a Noldo)
As Galadriel was one of the leaders of the Noldor leaving Valinor, she wasn't allowed to return yet to Valinor. But because she resisted the temptation of the Ring, because she refused to take the Ring which Frodo so willingly offered to her, the Valar granted her permission to return to Valinor. That's what she meant when she said 'i have passed the test' and 'i will diminish and go into the West'. With diminishing she means that she will allow the One Ring to be destroyed, so that also the Three Elven rings (she was bearer of Nenya, which is one of the Three Rings) would lose their power and the Elvish works/realms in Middle-Earth would fade.
So, that's about it i think. I hope it helps.
Give up the Halfring, she-elf...