The 40K wiki has quite a bit to say on the topic:
Before The Fall
The Eldar were a technologically advanced race, generally considered the most powerful race in the entire Galaxy. Their ancient empire spanned the Galaxy, and the only race that could have challenged them lay in hibernation. They had unparalleled mastery of the warp, so much so that they could even mold the mystical energies into physical materials. When mankind was still millennia away from building the Egyptian pyramids, the Eldar already ruled a vast, unchallengeable interstellar empire. The end of this empire came not from the actions of another race, but literally from within the Eldar themselves.
Their technology had advanced so far that little or no work was required by individual Eldar. As a result, at some point around the 24th or 25th millennia, groups of Eldar began forming cults dedicated to the pursuit of experiencing everything that life had to offer. These pursuits became more extreme and depraved as time went on. Torture and murder came to be seen as art forms every bit as worthy as poetry, music and painting. The Eldar were extremely long-lived (even immortal before the curse of the Nightbringer), never suffered from senility or disease, and their superior intellect allowed them to perfect their skills to a degree far beyond that of even the most talented humans. The Eldar were able to devise cunning new ways of torturing and killing their fellows at rates never imagined.
Government within the Eldar empire soon collapsed and the degeneration of their homeworlds and colonies continued unimpeded. As the pursuit of ever more extreme experiences reached its height, death reigned in the streets of Eldar cities, hunter and hunted each being part of a twisted ritual of destruction which consumed thousands. Some Eldar were able to see that their now-corrupt society was destroying itself, and fled in disgust; these refugees would settle in the distant colonies of their Empire, and would later be known as the Exodites.
[edit] The Fall
Upon dying, the soul of the deceased traverses the bounds of the physical realm and goes to rest within the warp. Generally, this has no effect on the physical realm. The Eldar, however, are quite different. The psychically-gifted Eldar have powerful souls and register a much stronger presence in the warp than do the vast majority of humanity.
The deaths of hundreds of thousands of Eldar in extremely pleasurable and/or painful ways -- and the concomitant entry of huge numbers of their souls into the warp -- had an unintended, and catastrophic, side-effect: The restless souls of the dead Eldar caused the formation of warp storms, which made travel extremely difficult in the physical realm. (This was one of the factors that precipitated humanity's Age of Strife.) A far worse effect, however, was that these souls began to somehow coalesce into a larger entity, a living representation of the corruption that had taken their lives. As more and more Eldar died to fuel the decadence and depravity of their society, this entity continued to grow, and the warp storms continued to grow worse.
In the early 30th millennium, this great warp entity finally attained sentience. In the last few years before its birth, many Eldar were driven insane by the uncontrolled energies possessed by this entity. Many Eldar who had stayed behind to try and turn their race away from the decadence that had consumed them saw what was happening and knew that something terrible was about to occur. These Eldar left the Empire, fleeing aboard colossal spacecraft called Craftworlds and taking any plants, animals or other pieces of Eldar culture with them.
At last the entity suddenly came to life, and was said to have "taken its first breath"; this first 'breath' instantly slew the overwhelming majority of the Eldar who lived in the empire as their souls were 'inhaled' by the entity. The death toll was so massive within the Eldar empire that warp energy literally spilled over into the physical realm, creating a permanent overlay of realspace and warpspace known as the Eye of Terror. Even the Eldar who had fled the Empire were affected -- many Exodites and Craftworlders fell dead as their souls were also sucked from them. As well, many psykers of other races were also driven insane or killed by the power that had been unleashed. This new entity became the fourth major Chaos god, Slaanesh, and the Eldar Empire was no more.
[edit] A New Chaos God and New Eldar Factions
Slaanesh's first act after annihilating so many Eldar was to hunt down their gods. Like they had with Slaanesh, the psychically gifted Eldar had previously unwittingly created their own pantheon of gods. With the sudden deaths of so many Eldar, the old gods were drastically weakened. Slaanesh took advantage of this weakness and destroyed them, assimilating them within herself (unlike other races, the Eldar consider Slaanesh to be female, and refer to her as "the Great Enemy" or "She Who Thirsts"). Ancient Eldar gods, such as Kurnous, Isha, Vaul, and Asuryan were hunted down and consumed by Slaanesh.
However, among the death and destruction, two Eldar gods survived. It is said that the god of war, Kaela Mensha Khaine, fought Slaanesh, and neither was able to defeat the other - Khaine, being a war god, could not be defeated in battle, but Slaanesh was far too powerful for Khaine to defeat. Some tellings of the story state that in the end, the Chaos god of war, Khorne, interceded on Khaine's behalf, likely with the aim of absorbing the Eldar god of war himself. Regardless of the actual events of the battle, the result is clear: Rather than be absorbed by the more powerful Chaos gods, Khaine's form was shattered and exiled to the mortal realm in the form of great statues called "Avatars", which rest in the hearts of the various Craftworlds.
The other Eldar god to survive the Fall was the trickster god Cegorach, also called the Laughing God. He is said to have hidden himself behind Khaine during the great duel with Slaanesh; when Khaine was finally banished, the Laughing God hid himself within the Webway, where he still hides. Occasionally he reenters the warp in order to play pranks upon Slaanesh, most famously to rescue the souls of certain Harlequins from consumption by her.
After Slaanesh's birth and her war with the Eldar gods, the few surviving Eldar found their society in ruins. They were split into factions, and were forced to find ways of protecting themselves from Slaanesh; with Slaanesh's existence so closely tied to the Eldar race, any Eldar who subsequently died were instantly consumed by the ever-hungry "Great Enemy". The new Eldar factions - the Craftworld Eldar, Dark Eldar, Harlequins, Outcasts, and Exodites - each found their own ways of dealing with the curse of Slaanesh.
[edit] Legacy
The Fall of the Eldar was a major event on the galactic stage. Firstly, the Eldar empire was wiped out, leaving behind the hell-region the Eye of Terror. The forces of Chaos gained another powerful member in the god Slaanesh, making them a further threat to the existence of mortals; however, Khorne's interference in the Slaanesh-Khaine duel caused a rift between Slaanesh and Khorne, and is one of the reasons why the two gods continue to be enemies. The Eldar went, virtually instantaneously, from being the most powerful force in the galaxy to being a few relatively small groups struggling to survive. The various Eldar factions are reviled by other races for their tendency to focus on their own survival, often at the expense of those other races.
The birth of Slaanesh had the side-effect of clearing the galaxy of the warp storms associated with the era prior to her birth. This allowed galactic travel to resume; the human leader known as The Emperor was able to take advantage of this to unify mankind under his command, end the Age of Strife, and begin his epic Great Crusade. The Great Crusade would see the forging of the galaxy-spanning Imperium, which would attempt to fill the power vacuum left by the near-annihilation of the Eldar.
Some Eldar did learn from their mistakes. Knowing that their psychic prowess has profound effects upon the warp, some far-sighted leaders among the Craftworld Eldar have noticed the slow growth of a new god, Ynnead, god of the dead, within the safe confines of the Eternal Matrix. The Eldar hope that if and when Ynnead is finally born, he will confront Slaanesh and defeat her once and for all. But as Eldrad Ulthran muses, this will take thousands of years at the very least, and his awakening will require the passing of every Eldar soul linked to the Eternal Matrix.
So yeah, as Mikey said they pretty much partied themselves to death. It's just that as the end drew near they were the sort of parties you probably wouldn't want to go to, what with all the torture and murder and stuff.
To expand on the various remaining factions of the Eldar:
Craftworld Eldar
These guys are basicaly the inheritors of the Eldar Empire. As it became more apparent that large quantities of excrement were about to hit the air conditioning unit, a load of Eldar piled whatever tech, animals and equipment they could find into massive starships known as Craftworlds and hauled ass out of there before Slaanesh came knocking. They have a culture very similar to the "classic" Eldar Empire's (before it went all loony, obviously) and maintain much of the last remnants of the Empire's technology and knowledge. They're one of the most advanced factions in the galaxy (second only to the Necrons, AFAIK), but are severely lacking in numbers due to their strict regulation of procreation. There are only about a dozen or so known Craftworlds, IIRC, and to all intents and purposes they function as completely seperate nations with little contact between each other. They mostly just drift around space aimlessly, feeling sorry for themselves.
Dark Eldar
The few survivors of the Eldar who followed the hedonistic lifestyle that led to the Fall make up what is known as the Dark Eldar. Since Slaanesh was going to have the survivors for desert, they all legged it into the Webway where they spend most of their time in hiding. They've got a city or planet of some sort (it's never quite made clear what is is) called Commoragh which they operate out of, and they continue to practice their sadistic and hedonistic ways. Like the Craftworld Eldar, they too maintain some good examples of Pre-Fall tech, but their numbers are incredibly limited and they've few resources, meaning they're nothing more than a particularly scary bunch of pirates.
Exodites
The Exodites are the guys who first saw that things were starting to go a little screwy, and decided to do a runner before stuff started going really wrong. They went off to the farthest reaches of the Eldar Empire and settled planets were they knew life would be hard, so as to prevent the hedonistic lifestyle springing up again. When the Fall came they were relatively untouched. They're generally considered rather backwards in comparison to most other Eldar factions (the Eldar version of the Amish, basically), though they still use and maintain tech that's at least on a par with that of the Imperium, and probably some stuff that's quite a bit more advanced. One of the Craftworlds, Biel-Tan, intends to use the Exodite worlds as the foundations for a new Eldar Empire sometime in the future.
Harlequins
The Harlequins are the smallest and least well known of the surviving Eldar factions. They're servants of their trickster god, and travel the Webway to both Eldar and non-Eldar settlements. They're sort of a combination of warriors and storytellers, and keep the mythology of the pre-Fall Eldar alive.
Outcasts
As the title suggest, they're those Eldar who have no real aliegence to any of the main groups. They're usually pirates, but other more pleasant groups are around as well.