According to Tolkien, in The Lord of the Rings' history and lore, only the Elves of Middle-earth and the beings called the Ainur, which refers to the Valar and their servants the Maiar, are immortal. Once Elves grow tired of their lives in Middle-earth, they are granted passage to the Undying Lands of Valinor, which is a paradise where life goes on forever.
Elves still die or can be killed, and in this case, they either dwell in the halls of Mandos, the Maia who is the caretaker of the dead, or are resurrected in Valinor. On the other hand, sometimes those of the mortal races earn a place on the ships departing from the Grey Havens.
5 Melkor
Also known as Morgoth, he's a Valar, and could never be destroyed or killed. His fate was to be judged by his peers in the Undying Lands, and his first lieutenant Sauron would have joined him, had he not fled to hide in Middle-earth and grow in strength again.
After Melkor had been soundly defeated at the end of the War of Wrath, he was brought back to the Valar, and this wouldn't be the first time that he would stand before them. His earlier transgressions in early history had been forgiven after he had done his penance in the halls of Mandos, but this time all that awaited him was eternal punishment and solitude. Melkor was cast into the Void where he could not harm the denizens of Eä ever again.
4 Elrond
Beren and Luthien had one child, Dior, and he had a daughter named Elwing. Her two sons were Elros and Elrond. Elrond was the brother to choose to live forever, and part of his motivation was to record the history of Middle-earth. He was born in the First Age and departs from the Grey Havens at the closing of the Third Age. His home of Rivendell became a repository of knowledge, and after he sailed west with Galadriel, her husband Celeborn took over.
Elrond is one of the younger immortals in the lore of The Lord of The Rings, being only about 2500 when Bilbo meets him in Rivendell. Healing and writing made Elrond famous, and he was so honored among his people that he married the daughter of Galadriel and had three children of his own, the youngest being Arwen Evenstar.
3 Gandalf
A Maia doesn't get to die unless their Vala permits them, or at least that seems to be the case with Gandalf. Originally known as Olórin, he was primarily the vassal of Manwë but served many of the Valar. It was said that his time with Nienna, the Valar of sorrow, was when he learned his most valuable lessons. It was pity, compassion, and understanding that led to Sauron's defeat, not the might of arms or magical power.
When he first came to Middle-earth, Olórin was close to other famous figures of the time. He and Glorfindel had been friends since the early days of Valinor, and his friend now resided in the Grey Havens. One of his neighbors, Cirdan the Shipwright, was also a friend of Olórin and held him in such high esteem that he entrusted him with one of the three Elven rings, Narya.
2 Galadriel
One of the oldest beings in Middle-earth by the time the War of the Ring had come to its conclusion, Galadriel saw the light of the Two Trees and knew Valinor before the attacks of Morgoth and Ungoliant. She was more than 10,000 years old when she boarded a ship leaving for the Undying Lands at the end of The Return of the King.
Galadriel had a special power to read the hearts and minds of people and judge them accordingly, a skill that made her a respected leader and wise in the ways of both Elves and mortals. She has held the same regard as her equally famous brother Finrod, one of the Elvish leaders who defeated Morgoth. It's possible that she refined and learned these powers from her close friend Melian, a powerful Maiar who had already departed by the time of the War of the Ring.
1 Gimli
Humans are not permitted to sail to Valinor, and the breaking of this ban is what sunk Numenor. Dwarves have even less interest in the Undying Lands, and Gimli of the Fellowship was the one that was ever allowed to sail there. He was already 242 years old when he left with his friend Legolas. It's possible that Gimli died in Valinor, as the Hobbits who were granted passage did, but there's no date given for his death, only his departure. It's nice to think that there might be one stubborn dwarf holding out among all those shining immortals.
LotR legend has it that the ship that carried Gimli and Legolas into the west was the last one to leave and that the Prince of the Greenwood built it himself. No doubt his arrival was also endorsed by Elrond, who formed the Fellowship of the Ring, and Galadriel, who had given him the priceless gift of three locks of her hair, shining with the light of the Two Trees.