For reference, the scene in question is on Youtube:
The line you reference
You're misremembering the line. From the script:
Galadriel (V.O.): The world is changed: I feel it in the water; I feel it in the earth; I smell it in the air. Much that once was is lost, for none now live who remember it. It began with the forging of the Great Rings.
Fellowship of the Ring (2001)
This line does not appear in the books quite like this.
The first part is taken from Treebeard, in Return of the King (bold emphasis is mine, italic is Tolkien's):
Then Treebeard said farewell to each of them in turn, and he bowed three times slowly and with great reverence to Celeborn and Galadriel. ‘It is long, long since we met by stock or by stone, A vanimar, vanimálion nostari!'1 he said. ‘It is sad that we should meet only thus at the ending. For the world is changing: I feel it in the water, I feel it in the earth, and I smell it in the air. I do not think we shall meet again.'
Return of the King Book VI Chapter 6: "Many Partings"
From the context, it's likely that the "change" Treebeard is referring to is the end of the Dominion of Elves in Middle-Earth. It's possible that this is also what Galadriel is talking about in the film, although we can't be sure.
It's worth noting that "the changing of the world" is an idea that comes up occasionally in Tolkien's Legendarium. Most often it refers to when Eru Ilúvatar turned the Earth into a sphere and took Aman beyond the reaches of Men, which became extremely significant to the idea of the Fading of the Elves and the diminishing of magic in the world.
The middle part, "Much that once was is lost..." is wholly an invention of the film. However, given the context, it doesn't make a lot of sense for Galadriel to be referring to the One Ring. The most likely interpretation is that this is meant to be addressing us in the present day; to me, this is how the movie is incorporating Tolkien's conceit that The Lord of the Rings is a translation of a long-forgotten mythic history.
Saying "None now live..." is then a bit of a simplification; from our perspective, Elrond and Galadriel aren't actually dead, but explaining the intricacies of Aman and Elvish theology to the average moviegoer is a tall order, even if Jackson had the rights to include that aspect of the story2.
The last part, "It began with the forging..." is obviously Galadriel's hook into setting up the backstory; what's not clear is what "it" is. My ongoing thesis is that "it" is the Fading of the Elves and Rise of Men, although from a meta perspective "it" could just as easily refer to "this story you are about to hear."
What's abundantly clear, however, is that Galadriel is not referring to the One Ring at this point.
The line you mean
I believe the line you're actually referring to is from later in the monologue:
Galadriel (V.O.): [T]he hearts of men are easily corrupted. And the Ring of Power has a will of its own. It betrayed Isildur to his death. And some things that should not have been forgotten...were lost. History became legend...legend became myth. And for two and a half thousand years the Ring passed out of all knowledge.
Fellowship of the Ring (2001)
This line is oddly phrased.
- The line "some things that should not have been forgotten..." doesn't appear to suggest that the Ring itself had been forgotten, but rather that nobody knew what had happened to it. Since it had been missing for so long, the Wise allowed themselves to believe it would never be recovered, which had the unfortunate consequence of them being totally blindsided by Sauron's re-emergence
As Joe C points out in comments, the line about "History became legend..." is a little hard to reconcile with Elrond's continued existence.
However, it's clear from the rest of the story that nobody really forgot about the One Ring; the Elves certainly didn't, although they allowed themselves to believe it was lost forever. Gondor didn't forget either, if Boromir's comment at the Council of Elrond is any indication (emphasis mine):
'[Elrond] was the herald of Gil-galad and marched with his host. I was at the Battle of Dagorlad before the Black Gate of Mordor, where we had the mastery: for the Spear of Gil-galad and the Sword of Elendil, Aiglos and Narsil, none could withstand. I beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own.'
At this the stranger, Boromir, broke in. 'So that is what became of the Ring!' he cried. 'If ever such a tale was told in the South, it has long been forgotten. I have heard of the Great Ring of him that we do not name; but we believed that it perished from the world in the ruin of his first realm. Isildur took it! That is tidings indeed.'
Fellowship of the Ring Book II Chapter 2: "The Council of Elrond"
Even in the film, nobody at the Council (Boromir least of all) seems remotely surprised to see that the Ring exists, merely that it was found.
I'm inclined to believe that Galadriel is being more poetic than literal here.
1 According to Letter 230:
Treebeard's greeting to Celeborn and Galadriel meant 'O beautiful ones, parents of beautiful children.'
The Letters of J.R.R. Tolkien 230: To Rhona Beare (Incomplete). June 1961
2 The specifics of what Aman means to the Elves isn't fully outlined until The Silmarillion, which Jackson does not have the film rights to. Much of the mythology had to be simplified for this reason