In The Lord of the Rings, much is made of the different ages that make up the history of Middle-earth. The events of the main series up to the conclusion of Return of the King happen at the very end of the Third Age, while the Last Alliance of Men and Elves brings the Second Age to a thunderous close (as depicted at the start of The Fellowship of the Ring.) The Silmarillion novel details the devastating war between the forces of good and Morgoth, the most terrible of the Valar.
Despite these three ages spanning over 6000 years, J.R.R. Tolkien's world of Arda has a far longer history. Though they were more loosely defined in terms of their start and end times, they featured some of the most important events and significant characters in the entire canon, some of whom survived to the time of The Lord of the Rings.
The Years of the Lamps Saw Morgoth's Emergence
After the creation of the universe, or Eä, the demigods of Tolkien's design, the Valar, head to Arda. The rogue Valar, Morgoth, claims it as his own and a devastating war between him and the rest of his kin leads to the world becoming "marred." This leads to the Elves' lives slowly fading while they reside there in later years. This effect is lessened to near zero in the land of Valinor, which is why all Elves must eventually make the trip there.
The age is known as the Years of the Lamps because of two light sources created by the good Valar and placed at the North and South of the world. It is also known as the days before days because none of the major races of Men, Elves or Dwarves yet walked the land. The end of this age is brought about once again by Morgoth, who manages to overthrow the lamps and escape without repercussions. After this happens, the Dark Lord dominates Middle-earth while the rest of his kind settle in Valinor to the West.
Elves Awaken in the Years of the Trees
The Years of the Trees span more than 14,300 years, with the last third overlapping with the First Age. It is named for the Two Trees of Valinor that the Valar once again created as a light source for Arda. The Elves, the chosen children of the god of Tolkien's universe Eru Ilúvatar, finally awaken and begin to settle across the world. The conflict between the Elves and Morgoth also begins in earnest, with several battles setting the stage for the great and terrible conflicts of the First Age.
Galadriel is born in the Y.T. 1362 and lives through the rest of this age. Understanding how long this is can be a little tricky to calculate, as Tolkien himself changed the conversion rate when trying to understand how long Valinorian years are in "normal" solar years. Though he initially chose 9.5 years for every year in Valinor, he altered this in the appendices of The Lord of the Rings to 144 years per 1 Valinor year. This would make her around 19,800 years old by the time the Years of the Trees end. The Trees themselves were destroyed by the great spider and ally of Morgoth, Ungoliant, and eventually replaced by the creation of the sun and the moon, bringing the age to a close.