Jagiello (alternatively Jogaila), the founder of the dynasty, was the Grand Duke of Lithuania, and also ruled Poland as [Wladyslaw II]? after marrying Jadwiga, queen of Poland. The ruling House of Jagiello, also known as the Jagiellonians, held the crown of Poland from his accession in 1386?, after the Polish House of Piast? died out.
While Jagiellonians were hereditary rulers of Lithuania, in Poland king was elected. So while Szlachta always elected Jagiellonian to save Polish-Lithuanian alliance, kings usually had to grant some new priviledges to be elected.
The Jagiellonian rulers (with dates of ruling Poland-Lithuania in brackets) were:
Sigismund's heir was his sister, Catherine, who married John III Vasa of Sweden; as a result, the main branch of the Jagiellonians merged with the House of Vasa.