In Canon Edit
Rivendell is the 'Last Homely House' between the inhabited regions of Eriador and the wilderland beyond the Misty Mountains. It sits beside the River Bruinen, in a deep valley cut through the foothills of the mountains. There is only one narrow bridge across the river, and this is only accessible by a steep path leading down from the moors above.
Known structures include Elrond's Hall, where feasts are held, and the Hall of Fire, where singing and storytelling take place. The two are across from each other. There are rooms on the ground floor (such as Bilbo's) and upstairs (where Frodo stays). Outside there are gardens. There must also be a forge, where the shards of Narsil are reforged into Anduril, and a stable to house visiting ponies and horses such as Asfaloth.
The Hobbit Edit
In The Hobbit, Gandalf, Bilbo, and the Dwarves make an unplanned stop in Rivendell to rest and to seek the advice of Elrond with regards to Thror's map of the Lonely Mountain. Elrond discovers moon-letters describing a secret entrance to the Mountain.
In the book, the company is treated with courtesy at all times, though the Dwarves are not entirely comfortable with the situation. The movieverse makes Elrond and Thorin are a little nastier to each other. It also features a meeting of the White Council, which involves weirdly flirtatious behavior on the part of Gandalf and Galadriel and gratuitous mustache-twirling from Saruman.
The Lord of the Rings Edit
Gandalf advises Frodo to take the Ring to Rivendell from the get-go. After much turmoil and a direct confrontation with the Black Riders at the Ford of Bruinen, he, Sam, Merry, Pippin, and Aragorn arrive in mid-October. (In the book, they are aided by Glorfindel; in the movie, Arwen shows up with Glorfindel's horse.) They rest there for several weeks while Frodo recovers from his injury and scouts are sent out to determine the fate of the Riders.
Meanwhile, the Council of Elrond takes place, and the decision is made to send the Ring to Mordor to be destroyed. A company is chosen for the quest: Frodo volunteers as the Ringbearer. Gandalf the Wizard and Aragorn the Ranger pledge to guide and aid him along the way, until the quest is complete. Boromir of Gondor, Legolas the Elf, and Gimli the Dwarf each go to represent their people, and are only sworn to go as far as their roads home take them. Sam, Merry, and Pippin volunteer out of friendship to Frodo, and are admitted to the company on Gandalf's advice, but against Elrond's better judgement. Thus, the Fellowship of the Ring totals nine, Nine Walkers representing the Free Peoples of Middle-earth set against the Nine Riders that are evil.
The Fellowship depart from Rivendell at dusk in late December.
In Badfic Edit
It is one of the regions of Arda most badly warped by Mary Sues, who often crash the Council, end up joining the Fellowship as Tenth Walkers, and force the company to depart at dawn instead of dusk. Although it is a house (admittedly, a rather large house), they will often refer to it as a city and call Elrond a king. Its name is commonly misspelled as 'Rivendale' in various fanfics by people who have only seen the Movies. (Or worse, Riverdale, by Archie comics fans.)