You may come across these factions during your adventures:
The Harpers are a semi-secret organization dedicated to promoting good, preserving history (including art and music of old) and maintaining a balance between civilization and nature by keeping kingdoms small and the destruction of plant life to a minimum. They consider the elven empire of Myth Drannor shortly before its fall to be the pinnacle of civilized history and strive to recreate the world in that image.
Those Who Harp are led by a council of seven High Harpers, who are responsible for most of the group's long-term plans and goals. High Harpers are elected through the means of secret ballots among the other High Harpers, with the criteria being long time service and extreme discretion in the implementation of their plans.
The Knights of Myth Drannor
The Knights of Myth Drannor adventuring company came to prominence as the rulers of Shadowdale with their successful defiance of Zhentil Keep. They were also instrumental in the defeat of Lashan of Scardale.
The Zhentarim, also known as the Black Network, are a mercenary company who, although based out of several strongholds, hold their centre of power in Zhentil Keep. They serve Bane, the God of Strife and Tyranny and look to further his influence and cause across the Realms. They are a highly dangerous organisation with a great deal of power and are the primary enemies of the Harpers.
The Red Wizards of Thay
The Red Wizards of Thay are an organization of wizards in Faerûn. They are the notorious and nefarious spellcasting ruling class of the inhospitable but well-inhabited country of Thay. Slavers, demonologists, and magical experimenters, the Red Wizards are much feared by most of Faerûn. They constantly scheme to bring down the neighbouring nation of Rashemen, where they are fought by the equally famed magicians the Witches of Rashemen.
A mysterious, subterranean race of elves, their skin is that of darkest night. They serve the Spider Queen Lolth, and are known to favour the usage of poisons on their weapons. They are renowned as both skilful warriors and dangerous spellcasters. They are truly deadly opponents. The Drow dwell in the Underdark, in large subterranean cities and ruled over by the Matron Mothers.
Note: this is by no means a comprehensive list of the factions involved in this, there are some I'm keeping aside by way of surprise.