Short answers, pulled from the story.
Babur was a direct descendant of Timur the great Turkic conqueror and Genghis Khan through the Chagatai Khanate. His dual heritage combined the Timurid and Mongol traditions to legitimize his rule over India.
Timur was born around the year 1336 and died in 1405. He rose to power by 1369 and ruled until his death in 1405.
Timur gained legitimacy by marrying Saray Mulk-khanum the widow of Husayn of Balkh in 1370. This union made him an in-law of a direct descendant of Genghis Khan and allowed him to style himself as Temür Gurgan or son-in-law of the Great Khan.
Babur traced his lineage directly to Genghis Khan via the Chagatai Khanate established by Chagatai Khan the second son of Genghis. The Tarikh-i-Rashidi documents this lineage with meticulous detail.
The Mughal Empire was founded by Babur in the early 16th century. It became one of the last great Mongol successor states carrying forward the traditions of both Timur and Genghis Khan.