After Sauron is defeated, as Gandalf and Aragorn are waiting for a sign, Gandalf utters the above line to Aragorn.
My question is, what is the line of Nimloth, and how it is related to the sign they are waiting for?
Nimloth, Sindarin for "white blossom", was the name of the White Tree of Númenor. Nimloth was a seedling of Celeborn, which was a seedling of Galathilion, which was created by Yavanna in the image of Telperion, one of the Two Trees of Valinor. Thus Nimloth's line dates back to Valinor. This is the tree that they're searching for (and found) a seedling of.
The full saying, on the finding of the tree, is Gandalf analogizing the hiding of Aragorn's line to that of Nimloth, the White Tree:
Here it has lain, hidden on the mountain, even as the race of Elendil lay hidden in the wastes of the North. Yet the line of Nimloth is older far than your line, King Elessar.'