Short answers, pulled from the story.
It was heralded by the Battle of the Kalka River on the 31st of May 1223, when Mongol forces defeated a combined Rus' army before withdrawing. The full-scale invasion under Batu Khan followed in 1236-1238, with Kiev falling in 1240, and the invasion ended with the Mongol succession process triggered by the death of Ögedei Khan.
Batu Khan led the invasion alongside the general Subutai, commanding a Mongolian Great Khanate army of around 120,000 mounted archers. Other Chingizid princes, including Orda, Güyük, Möngke, Kulkan, Kadan, and Büri, commanded forces during individual sieges such as Ryazan.
Kiev, which held about 50,000 inhabitants before the invasion, was besieged and stormed by the Mongols in December 1240 after the approach began in November of that year. The city was pillaged extensively, though the building of St. Sophia Cathedral survived intact, and according to the papal envoy Giovanni da Pian del Carpine, only about two hundred houses remained standing when he passed through in 1246.
Mongol demands for tribute from Russian princes continued until about 1480, even though a Russian army had already defeated the Mongols at the Battle of Kulikovo in 1380. Principalities that avoided physical conquest, such as Galicia-Volhynia, Polotsk, and Novgorod, were still forced to accept Mongol supremacy through tribute.
Galicia-Volhynia, Pskov, Smolensk, Polotsk, and Vitebsk avoided complete destruction or suffered little to no damage during the invasion. Novgorod was spared after its principality chose to preemptively surrender rather than face the Mongol advance.
The invasion was ended by the Mongol succession process that followed the death of Ögedei Khan. Heavy losses the Mongols suffered during the campaign also significantly weakened their subsequent operations, which helped spare the Holy Roman Empire and other western countries from further conquest.