A name given by J.R.R. Tolkien to Telperion, the Eldest of Trees, which grew in Valinor before the First Age. Each of Telperion's descendants is also called 'the White Tree' in its time.
When the Eldar first settled in the Uttermost West during the Chaining of Melkor, Telperion became the thing they loved the most in all Valinor. And so,
Yavanna made for them a tree like to the lesser image of Telperion, save that it did not give light of its own being; Galathilion it was named in the Sindarin tongue.1
Galathilion grew in the center of Tirion upon Túna where the Vanyar and Noldor dwelt. After the destruction of the Two Trees, the White Tree becomes a manifestation of the power of the Valar, and a symbol of the faith of the Eldar, and later the Dúnedain, to the Valar and to Ilúvatar. After the Noldor were exiled from Aman, they made images of Galathilion as a reminder of what they had been exiled from. The western gates of Moria were wrought by the smiths of Eregion to include such an image. In Lothlórien, we encounter an image of Galthilion wrought in silver in the house of Celeborn and Galadriel.
In the meantime, Celeborn, a seedling of Galathilion, was given to the Teleri, who planted it in the midst of Tol Eressëa. A seedling of Celeborn was one of the many gifts the Teleri brought to Númenor. Tolkien tells us:
...the tree grew and blossomed in the courts of the King in Armenelos; Nimloth it was named, and flowered in the evening, and the shadows of night it filled with its fragrance.2
Nimloth's blossoms open as Anar wanes, a reminder of Telperion's old cycle with Laurelin. Instead of a mingling of the lights, we have a mingling of the dimming Sun and Nimloth's fragrance.
Nimloth was destroyed by the last King of Númenor, Ar-Pharazôn the Golden, after the latter gave himself wholly to his erstwhile prisoner, Sauron. In the destruction of Nimloth, Tolkien demonstrates for us how irredeemably far the Númenorians fallen, and portends the awful destruction to come. The tree was burned in the temple of Ilúvatar on Meneltarma, an act of blasphemy I've always considered the true trigger of the calamity, beyond even Ar-Pharazôn's invasion of the Undying Lands.
If the destruction of Nimloth was Tolkein's way of telling us that matters in Númenor were irretrievable, we are still left with a glimmer of hope. Elendil's son Isildur was able to sneak into the courts of Armenelos and steal a fruit of Nimloth before her destruction.
Tall ships and tall kings
Three times three,
What brought they from the foundered land
Over the flowing sea?
Seven stars and seven stones
and one White Tree.3
The White Tree was one of the manifestations of the Uttermost West that the Faithful brought with them to Middle-Earth; indeed, the seedling was said to have protected Isildur's ship during its voyage. When Isildur and his brother Anárion founded the realm of Gondor, the White Tree was planted in Isildur's city of Minas Ithil. Images of the White Tree became the principal symbol of Gondor.
Anárion was slain during the Battle of Mount Doom, where Elendil and Gil-Galad overthrew Sauron before perishing themselves. In memory of his brother, Isildur planted a White Tree in Minas Anor. Tolkien does not specify whether Isildur transplanted the tree from Minas Ithil, or planted a seedling from it. If it is the latter, the tree was certainly destroyed when Minas Ithil fell into shadow and became Minas Morgul.
Whatever the case, the health of the White Tree always mirrored the strength of Gondor. The tree in Minas Anor flourished until TA 1636, when it died during the Great Plague4. King Tarondor replaced it with a new seedling in TA 1640, when he moved the capital of Gondor there from Osgiliath. This tree lasted until TA 2852, when it died at the same time as Steward Belecthor II.
Since seedlings of the tree could not be found, the dead tree was left standing in Minas Tirith (as Minas Anor came to be called), a symbol of Gondor's decrepitude.
This situation lasted for 167 years, until after the downfall of Sauron, as depicted in The Lord of the Rings. Immediately after his coronation, King Elessar discovered a seedling of the White Tree while wandering with Gandalf on Amon Mindolluin. Thus Gondor began the Fourth Age anew with a living 'scion of the Eldest of Trees'.
1The Silmarillion, first American edition (1977), p. 59
2Ibid ("Akallabêth") p. 263
3The Two Towers, chapter 11, 'The Palantir'.
4The Return of the King, Appendix B, 'The Tale of Years'.