Short answers, pulled from the story.
The name Rivendell translates directly from the Sindarin word Imladris, which means deep valley. Tolkien constructed the English name from riven meaning split or cloven and dell referring to a small valley.
Elrond founded Rivendell during the Second Age following the destruction of the Elvish land of Eregion by Sauron. Gil-galad gave him the Ring Vilya to protect the settlement and slow time within its boundaries.
Tolkien based the physical description of Rivendell on his 1911 hiking trip to Lauterbrunnen in Switzerland. The latitude matched that of Oxford where Tolkien worked while the river Bruinen flowed through the fictional location near the Misty Mountains.
The Fellowship set out on its quest on the 25th of December from Rivendell according to Anglo-Saxon tradition. They succeeded in destroying the Ring on the 25th of March as noted by scholar Tom Shippey regarding Tolkien's careful use of dates.
Peter Jackson represented Rivendell in 2001 using Kaitoke Regional Park in New Zealand with waterfalls added via computer-generated imagery. The design featured natural motifs suggesting integration with nature yet at one remove compared to Tolkien's nature-close versions.