Short answers, pulled from the story.
The Russian conquest of the Caucasus mainly occurred between 1800 and 1864. Russia annexed eastern Georgia in 1800, and the last regions were brought under Russian control by 1864, when the Russo-Circassian War ended with the expulsion of several hundred thousand Circassians to the Ottoman Empire.
The Caucasian War was the name given to the difficult conquest of the mountains between the Black Sea and Caspian Sea. It encompassed multiple conflicts, including the Murid War in Dagestan and Chechnya, which lasted from 1830 to 1859, and the Russo-Circassian War in the western mountains, which ended in 1864.
Pavel Tsitsianov served as commander-in-chief of Georgia from September 1802 and rapidly expanded Russian control from the Black Sea to the Caspian Sea by 1806. He subdued the Djaro-Belokani area, conquered the Ganja Khanate in January 1804, and secured the submission of the Karabakh, Shaki, and Shirvan khanates before he was killed outside Baku in early 1806.
At the end of the Russo-Circassian War in 1864, several hundred thousand Circassians were expelled to the Ottoman Empire and their lands were settled by Cossacks. The Principality of Abkhazia was also formally abolished that same year.
The conquest gave rise to a body of work known as the Literary Caucasus, which included writings by Alexander Pushkin, Mikhail Lermontov, and Leo Tolstoy. Pushkin's poem "The Prisoner of the Caucasus," written between 1820 and 1822, and Tolstoy's "Hadji Murat," published in 1912, are among the most notable works in this tradition.
The Treaty of Gulistan was signed in 1813, ending the Russo-Persian War of 1804-1813. Under its terms, Persia formally recognized Russian possession of the khanates of eastern Transcaucasia and the Caspian coast, including modern-day southern Dagestan, eastern Georgia, and most of what is now Azerbaijan.