Short answers, pulled from the story.
Roman general Pompey captured Jerusalem in 63 BC, which marked the end of independent Hasmonean rule. He intervened in a civil war between Hyrcanus II and Aristobulus II before appointing Hyrcanus II as ethnarch and High Priest.
Augustus appointed Publius Sulpicius Quirinius as Legate of Syria to conduct a tax census of Syria and Judea in 6 CE. Coponius served as the first prefect of Judea following this event and inherited Herodian military forces consisting of one cavalry unit and five infantry cohorts.
Pontius Pilate served as Roman prefect between 30 and 33 CE and crucified Jesus of Nazareth on charges of sedition. His brutality caused Samaritans to complain to Lucius Vitellius the Elder, who removed Pilate from his post and sent him to Rome.
Titus laid a five-month siege to Jerusalem in 70 CE with forces totaling around 50,000 soldiers. The Romans breached the walls one by one and stormed the Temple Mount destroying the Second Temple.
Simon bar Kokhba led the Bar Kokhba revolt which erupted in 132 CE because Emperor Hadrian established Aelia Capitolina atop the ruins of Jerusalem. This pagan Roman colony was dedicated to Jupiter and precipitated the final major Jewish revolt against Roman rule.