Long before the beginning of the First Age, the Elves of the Great Journey travelled westward through the lands of Middle-earth. Coming to the Great River Anduin and the high peaks of the Misty Mountains, some of the Elves of the clan of the Teleri fell away from the journey, and settled in the woodlands east of the Mountains. These were the original Silvan Elves, who lived on either side of the River. At this time in their history, all of this people still lived close together, with some dwelling in the land that would later be called Lórien, and the others settling around the hill of Amon Lanc in the far south of Greenwood the Great.
It must have been in the time these Elves were living closely together that the Silvan Elvish language appeared. As history passed, the Elves would move away from one another, and mingle with other Elvish peoples, so that the Silvan branch of Elvish would eventually become extinct. Nonetheless, relics of that ancient tongue survived in some well-known place-names and personal names, such as Caras Galadhon, Amroth and even Lórien itself.
The Silvan Elves dwelt in their twin woodland realms for many centuries, but in the Second Age the emerging power of Sauron began to drive them apart. Oropher was the ruler of the Elves who dwelt in the Greenwood, and he began to seek safety by moving his people northwards, away from Amon Lanc and away from the Silvan Elves who lived to the west of the Great River.
West of Anduin: the Silvan Elves of Lórien
Unlike their cousins in the east, the Silvan Elves of Lórien remained settled in their land throughout their known history. We know little of their existence well into the Second Age, where we find them ruled by a Sindarin Elf named Amdír. After the destruction of Eregion in the middle Second Age, many of its exiles had escaped through Khazad-dûm into the east, and had joined the Silvan Elves of Lórien and ultimately mingled with them.
Amdír was lost in the wars at the end of the Second Age. In the time of his son Amroth, Durin's Bane was unleashed in Khazad-dûm. After this time, especially with the power of Sauron growing in Dol Guldur to the east, many of Lórien's people fled into the south. Amroth himself was among them, and he was drowned in the Bay of Belfalas far from his kingdom. It was at this time, a little more than a thousand years before the War of the Ring, that Galadriel and Celeborn became the Lord and Lady of Lórien.
East of Anduin: the Silvan Elves of Mirkwood
The Elves of Greenwood the Great, soon to become known as Mirkwood, had a much less settled existence than their fellow Silvan Elves in the west. Their original dwellings were in the far south of the great Wood, but as Sauron's power grew, they gradually withdrew farther and farther northward. Eventually Sauron chose to settle within the Forest itself, and in fact built his fortress of Dol Guldur on the hill of Amon Lanc, where the Silvan Elves had originally dwelt. Though his identity remained secret during this time (he was known simply as the 'Necromancer') he brought darkness and shadow to Greenwood the Great, and the Elves retreated farther still.
In the later years of the Third Age, their King Thranduil (the son of Oropher, who was slain in the War of the Last Alliance) dwelt with his people in the far northeast of Mirkwood, in underground dwellings protected by magic. Their history of fleeing from Sauron's darkness had made them a suspicious and unwelcoming people, as Thorin and his companions discovered during the Quest of Erebor. It was from Thranduil's underground halls that his son Legolas journeyed at the end of the Third Age, to play a great part in the War of the Ring.
The War of the Ring and beyond
In the War of the Ring, Sauron's forces attacked both Silvan realms. By the power of Galadriel and her Ring, Lórien was protected against three waves of assault from Dol Guldur. Thranduil had no Ring to protect his realm - it was invaded and the trees of the northern forest set alight, though he was at last able to drive back the enemy.
With the fall of Sauron, Dol Guldur was destroyed, and Mirkwood renamed again: Eryn Lasgalen, the Wood of Greenleaves. Now the Silvan Elves once again dwelt in the south of the Wood, but these were not Thranduil's people, but Elves of Lórien, who crossed the Great River to live in a new land named East Lórien. Though the original land of Lórien faded after the passing of Galadriel and Celeborn, the Silvan Elves of the Wood lived peacefully under their trees for many long years.