(This is about the Eldar in the Games Workshop's Warhammer 40K universe, not Tolkien's :))
The Fall occurred approximately ten thousand years ago, around the 31st millennium. It is not categorically known what exactly happened, but the Eldar have stories and legends about it. What follows is the story of the Fall as is generally accepted by the Eldar.
Before the Fall, the Eldar civilization was at its height. Their worlds were beautiful, peaceful places, filled with wonders. The Eldar were technically superior to all other races in the galaxy, and viewed them as barbarians.
Back then, there was no such thing as Craftworlds or the Path. All Eldar investigated every curiosity, indulged every whim, as the fancy took them. Many wonderful things were created and learned, things that are now forgotten. The Eldar lived long lives, and when they died their spirits dissolved peacefully back into the Warp (there was no need for spirit stones as the warp was not filled with the horrors that exist today).
The pride of the Eldar proved to be their downfall. They believed all secrets were theirs to discover, all pleasures theirs to partake. Exotic cults sprang up everywhere, dedicated to all types of sensual excesses or esoteric knowledge. The corruption turned quickly to wanton abandon. Blood flowed through the streets as the Eldar sought more and more perverse and murderous pleasures.
Within the Warp, the destruction of the Eldar race took shape. It was here that the departed spirits of dead Eldar began to coalesce into a singular consciousness. It was a consciousness made in the image of the Eldar, a shadow of themselves, of what they had become. This consciousness was to become the Chaos God Slaanesh.
The Eldar were riven with madness for years as Slaanesh dreamed in the warp. When Slaanesh was born, there wasn't a single Eldar anywhere who didn't feel the agony. A psychic implosion ripped through the universe. The spirits of the Eldar were ripped from their bodies and consumed by the Chaos God.
The centre of this massive implosion lay at the heart of the Eldar worlds. All Eldar within thousands of light years were killed instantly, spirits drawn into the warp to feed the hungry god. Most of the Craftworlds were destroyed and even the Exodites were affected, leaving only the most remote of them untouched.
The psychic shockwave focused on the Eldar mind, but millions of humans and creatures from other races were destroyed also. The Warp was engulfed in turmoil for days. The very fabric of space itself was rent apart, and the Warp spilled into the material universe. This hole in space spread until it encompassed the Eldar realms of old and reached the limits of its power. Nowadays this is known as the Eye of Terror, and it is the largest zone of its kind in the galaxy. Here the warp and material universe overlap. Chaos Space Marines rule over planets of blood and fire and plot their invasions against the hatred Imperium. Daemons gibber and bathe in the energy of the warp.
The only positive side to the birth of Slaanesh was that the Warp was thereafter becalmed. Before, the Warp was riven with storms and fierce tempests, making it all but impossible for starships to travel through. Now it became passive. Human starships flew from Earth for the first time in thousands of years, and brought into contact the Human worlds of old. The Emperor led the Great Crusade, uniting mankind into the Imperium, and mankind thus replaced the Eldar as the galaxy's primary race.
This information was gleaned from the Second Edition Codex Eldar (the 3rd edition one doesn't have anywhere near as much background information, sadly).
To those who downvoted me and softlinked to the copyright notes: this is my own words. I didn't copy&paste it, thankyouverymuch. :P