The Noldor were numerous and powerful, yet by the time of the events of The Lord of the Rings they are few and far between. Why did this once great group of Elves all but vanish by then?
In short, pride and the mistake of following Fëanor. Also, they only "vanished" from Middle-Earth.
The Noldor were the Second Clan, and thus second in size to the Vanyar. They were crafty and proud, leading to the defining events of the Silmarillion. Fëanor rebelled against his brother Fingolfin, and was banished; his brother remained High King of the Noldor. Then, Fëanor created the Silmarils, which made Melkor, the original bad-guy, extremely envious. He stole them and fled to Beleriand, upon which Fëanor swore an oath to get them back. He pursued Melkor, and the Noldor were divided. Although they held Fingolfin to be their actual High King, the greater majority of the Noldor followed Fëanor across the sea. Some remained in Valinor under Fingolfin.
In order to leave Valinor, the Noldor asked to use the ships of the Teleri. The Teleri elves refused, and the Noldor elves under Fëanor committed the first Kinslaying. A messenger of the Valar pronounced a Doom upon them, that if they should continue they would be unsuccessful in recovering the Silmarils, and they would all die or be tormented by grief. Some of the group who committed the Kinslaying repented and returned to the Valar, who forgave them and allowed them to remain in Valinor. Of the remaining group: a small portion, led by Fëanor, abandoned the greater host who followed Fongolfin by sailing across the sea in their stolen ships, which they burned upon arrival (to the dismay and objection of some who wanted to return to ferry the remainder across).
Unfortunately, this started a feud in the Royal Houses of the Noldor. When Fingolfin heard about all this, he took the only possible route to Middle-Earth - going north and crossing the Helcaraxë, or the Grinding Ice. This was a bad move and killed a lot of the Noldor who had stayed with Fingolfin. Shortly thereafter, Fëanor was killed after being attacked by Balrogs, and the feud between the Royal Houses was settled.
The Noldor hung around Beleriand for a while, embroiled in a war with Melkor/Morgoth and creating alliances with the Sindar elves and the first tribes of Men. After Fëanor's death, Fingolfin reigned again as High King in Hithlum, and his son Turgon built Gondolin, the hidden city. Nargothrond also belonged to them. They fought multiple battles against Morgoth, including the Dagor Aglareb and the Siege of Angband and the Battle of Sudden Flame, when Morgoth broke the siege. Fingolfin challenged Morgoth to single combat and though he dealt him many wounds, was killed. His son Fingon became High King.
The upshot of all this is that it led to the Nirnaeth Arnoediad, the Battle of Unnumbered Tears. The Noldor were betrayed by the Easterings and utterly defeated, along with their allies. They were scattered, as their strongholds continued to fall. Morgoth defeated Nargothrond soon after, and Gondolin was betrayed. Turgon managed to help many of his people escape, but had no heirs himself, so the last son of Fingolfin, Gil-galad, became High King. Very soon after this, the Valar came to Middle-Earth to deal with Melkor/Morgoth and cast him out, sinking Beleriand in the process.
Most of the Noldor sailed back to Valinor after this, although some remained in Middle-Earth for various reasons. Galadriel and Celebrimbor refused the Valar's pardon and remained. Gil-galad founded his kingdom in Middle-Earth and reigned during the Second Age. When Gil-galad perished during the Last Alliance, the High Kingship of the Noldor passed out of Middle-Earth and went to Finarfin, who was among the Noldor that had never left Valinor.
So by the time of the events of the Lord of the Rings, the reason that there are so few Noldor around is that most of them have gone back to Valinor after being pardoned. Indeed the last elves to leave Middle-Earth are Noldor (Galadriel and Elrond). Yes, a lot of them were killed in the wars with Morgoth, but they didn't so much vanish as sail back across the sea to be reunited with the rest of their kin.
Disclaimer: I haven't read the Silmarillion in a while, so I used the Wiki page to help me make sure I had events in the appropriate order and names correct (although I remembered some of the strange ones perfectly, like Helcaraxë) See WP:Noldor.
*Also, remember that elves don't die in the traditional sense. They go to the Halls of Mandos and come back after a time if I remember correctly. So once again, they're in Valinor and not necessarily vanished.
Not all the Noldor returned to Middle-Earth with Fëanor. Of those that did, most were killed in the sackings of Gondolin and Nargothrond, and in the great battles against Morgoth, especially the Nírnaeth Arnoediad and the final War of Wrath.
Edited As requested by Keen, here's some more explanation. This is all from the Silmarillion and Unfinished Tales, although I haven't got them in front of me to give exact references.
In the First Age, Fëanor and his sons rebelled against the Valar (basically, the Gods) and returned to Middle-Earth in pursuit of Morgoth, who had stolen the Silmarils - great jewels that Fëanor had made. Many of the Noldor followed them. However, the initial battles were of mixed success and Fëanor was killed, leading the various populations led by his sons to eventually retreat into fortified cities. One of these was Nargothrond, which was an underground city by a river - heavily fortified and hidden, although its general area was known. The other was Gondolin, whose location was a complete mystery - it was in a hidden mountain vale, the people did not leave, and anyone who found it by accident was forced to remain.
Various battles followed over the next couple of hundred years, the most devastating of which was the Nírnaeth Arnoediad, the Battle of Unnumbered Tears. Later, Nargothrond was destroyed, thanks to the rash actions of Turín Turambar and the betrayal caused by the dragon Glaurung. Separately, Gondolin was destroyed, betrayed by Maeglin, nephew of its king, Turgon.
The First Age ended with the climactic War of Wrath, in which the Valar finally came to the aid of the Elves and Men. Morgoth was defeated, but at great cost, both in lives and in damage: in fact most of the land of Beleriand, where all these events took place, sank beneath the sea.
This is covered in The Silmarillion.
The Noldor all went to Valinor in the first age. Some returned (following Fëanor) to regain the Silmarils from Morgoth and in the ensuing wars many of these died. After the first age many returned; by the time of Lord of the Rings (third age) only the last few remain.
There is a decent summary of the history of the Noldor on Wikipedia: http://en.wikipedia.org/wiki/Noldor
When Orome found the Elves at the awakening site, most of them moved to Valinor and dwelt there. In total, there were 56 Noldor Elves awakened, and this number probably grew before the group left for Valinor.
And so it was that the Quendi ever after reckoned in twelves, and that 144 was for long their highest number, so that in none of their later tongues was there any common name for a greater number. And so also it came about that the 'Companions of Imin' or the Eldest Company (of whom came the Vanyar) were nonetheless only fourteen in all, and the smallest company; and the 'Companions of Tata' (of whom came the Noldor) were fifty-six in all; but the 'Companions of Enel' although the Youngest Company were the largest; from them came the Teleri (or Lindar), and they were in the beginning seventy-four in all.
Afterwards, following the theft of the Silmarils by Melkor; Feanor, then High King of the Noldor (after his father Finwe was slain), convinced a large number of the Noldor in Valinor to seek out Morgoth(no longer Melkor) in Middle-Earth. Thus they return from Valinor back to Middle-Earth: known as the Return of the Noldor.
The first group is under Feanor's command, second under his second brother Fingolfin's and last under his third brother Finarfin. Amongst the three groups, Finarfin's was the most reluctant. Most of Finarfin's group turned back to Valinor after the 1st Kinslaying took place. The hosts under Feanor and Fingolfin and a remnant of Finarfin's went on.
Many of the Noldor (including all the Noldorian princes except Gil-Galad) who went to Middle-Earth died throughout the pre-events of the War of the Great Jewels and finally the War of Wrath itself.
In Dagor-nuin-Giliath, Feanor fell. Fingolfin's son Argon and a number of Noldor also fell in the ensuing Battle of the Lammoth.
One of the greatest losses for the Noldor came during Dagor Bragollach, when the siege of Angband ended. This lead to the deaths of many Noldor in Beleriand and death of High King Fingolfin, being given the title of Feanor's son Maedhros was rescued.
After which is the events of the Nírnaeth Arnoediad. Even more Noldor died then, including High King Fingon, son of Fingolfin. The Kingship then passed to his brother Turgon.
Around this time came the fall of Hithlum and Nargothand; Kingdoms of the Noldor elves. Finally, the only Noldorin kingdom left was Gondolin. This too was destroyed because of a betrayal, and the Noldor were greatly diminished.
A small group escaped to the Mouths of Sirion, however. Not long after, the remaining Sons of Feanor learnt that Elwing daughter of Dior had a Silmaril in her keeping, and they committed the 3rd Kinslaying.
Hence, by the time of the War of Wrath, the Noldor in Middle-Earth were nearly annihilated. Only Gil-galad of all the Noldorin princes had survived.
Hence, quite a number of the Noldor who were left left for Valinor at the end of the war and First Age. The remaining few became founded their own lands (Gil-Galad, Galadriel).
Yet not all the Eldalië were willing to forsake the Hither Lands where they had long suffered and long dwelt; and some lingered many an age in Middle-earth. Among those were Círdan the Shipwright, and Celeborn of Doriath, with Galadriel his wife, who alone remained of those who led the Noldor to exile in Beleriand. In Middle-earth dwelt also Gil-galad the High King, and with him was Elrond Half-elven, who chose, as was granted to him, to be numbered among the Eldar; but Elros his brother chose to abide with Men. And from these brethren alone has come among Men the blood of the Firstborn and a strain of the spirits divine that were before Arda; for they were the sons of Elwing, Dior's daughter, Lúthien's son, child of Thingol and Melian; and Eärendil their father was the son of Idril Celebrindal, Turgon's daughter of Gondolin.
The Silmarillion - Of Earendil and the War of Wrath
By the end of the Third Age, very little of the Noldor remain as over the Second and Third Ages, the ones that remained in Middle-Earth at the end of the First Age slowly left for Valinor. Gildor Inglorian and Galadriel are one of the last Noldor to leave Middle-Earth.
There was Gildor and many fair Elven folk; and there to Sam's wonder rode Elrond and Galadriel.