Short answers, pulled from the story.
The Second Chechen War began on the 7th of August 1999, when Shamil Basayev and Arab warlord Ibn al-Khattab led up to 2,000 fighters from Chechnya into the neighboring Russian republic of Dagestan. A subsequent series of apartment bombings in Moscow and other Russian cities killed over 350 people, and Prime Minister Vladimir Putin ordered a military campaign against Chechnya in response.
The Russian siege of Grozny began in early December 1999 and ended on the 2nd of February 2000, when the Russian army seized the city. Russian Defense Minister Igor Sergeyev reported that at least 1,500 separatists were killed attempting to leave. The siege and bombardment killed between 5,000 and 8,000 civilians, and in 2003 the United Nations called Grozny the most destroyed city on Earth since World War II.
The principal separatist leaders included elected president Aslan Maskhadov, who was killed by Russian security forces on the 8th of March 2005, and warlord Shamil Basayev, who directed many of the major terrorist attacks and was killed in July 2006. After their deaths, Doku Umarov assumed leadership and in October 2007 proclaimed the Caucasus Emirate, shifting the movement toward Islamism before he was killed by poisoning in 2013.
Casualty figures vary widely and are impossible to verify independently. According to Amnesty International in 2007, the second war killed up to 25,000 civilians since 1999, with up to another 5,000 people missing. Western European rights groups estimated approximately 5,000 forced disappearances in Chechnya since 1999, and a count by the Russian human rights group Memorial in 2007 put the figure of civilians who died or disappeared since 1999 at up to 25,000.
The Beslan school hostage crisis occurred in 2004 as part of a campaign of terrorism directed mostly by Shamil Basayev. At least 334 people died in the attack. It was one of 23 Chechen-related suicide attacks that took place between June 2000 and September 2004.
On the 16th of April 2009, FSB head Alexander Bortnikov officially announced that Russia had ended its counter-terrorism operation in Chechnya. As of 2009, close to 480 active insurgents were still fighting in the mountains under the leadership of Doku Umarov according to official data, and clashes with militants continued in neighboring Dagestan and Ingushetia.