Short answers, pulled from the story.
The Jagiellonian dynasty was founded by Jogaila, Grand Duke of Lithuania, who converted to Christianity and was baptized as Ladislaus in 1386. He then married the twelve-year-old Queen Hedwig of Poland and became King of Poland, transforming his branch of the Lithuanian Gediminid dynasty into a royal house.
The Jagiellons ruled as Kings of Poland from 1386 to 1572, Grand Dukes of Lithuania from 1377 to 1572 (with interruptions), Kings of Hungary from 1440 to 1526, and Kings of Bohemia from 1471 to 1526. At their peak, members of the dynasty simultaneously held thrones across Poland, Lithuania, Hungary, and Bohemia.
The male line of the Jagiellonian dynasty ended with Sigismund II Augustus, who died in 1572 without any living children from his three marriages. The female line continued to rule Poland and Lithuania as elective monarchs until 1668, merging through Catherine Jagiellon's marriage into the House of Vasa.
On the 29th of August 1526, the Jagiellonian king Louis II of Hungary led his forces against Suleiman the Magnificent at Mohács. The Hungarian army was surrounded in a pincer movement and destroyed by Ottoman cavalry and Janissary musketeers. The twenty-year-old Louis died in a marsh during the retreat, ending the Jagiellonian line in Hungary and Bohemia.
The Union of Lublin, signed in 1569, merged the Kingdom of Poland and the Grand Duchy of Lithuania into the Polish-Lithuanian Commonwealth, also called the Republic of the Two Nations. Sigismund II Augustus pursued the union because of the costly and unsuccessful Livonian War and because Lithuanian nobility demanded equal rights with Polish szlachta. As he had no heirs, the union was his way of securing a shared political future for both states.
The Jagiellonian dynasty's legacy includes the Jagiellonian University in Kraków, the Jagiellonian Library, and the Globus Jagellonicus, which some scholars consider the oldest existing globe to depict the Americas. The dynasty is also associated with the Polish Golden Age, the period under Sigismund I and Sigismund II when Renaissance culture flourished and illiteracy fell as nearly every parish ran a school by the end of the 16th century.