Short answers, pulled from the story.
The Muslim conquest of Persia was the Rashidun Caliphate's conquest of the Sasanian Empire between 633 and 651, part of the early Muslim conquests. It ended the Sasanian dynasty and brought Persia under Muslim rule through sustained campaigns, sieges, and the suppression of rebellions.
The conquest took place between 633 and 651. A first Rashidun invasion began in 633, a second decisive invasion began in 636 with the Battle of al-Qadisiyyah, and Caliph Umar ordered a full-scale invasion of the rest of the empire in 642 after the Battle of Nahavand.
The Sasanian Empire had been drained by the Byzantine-Sasanian War of 602 to 628 and destabilized by the execution of Khosrow II in 628, which produced ten royal claimants in four years and a civil war. Parvaneh Pourshariati argues the empire was a decentralized confederation with the Parthians, who withdrew and made peace with the Arabs, leaving the Sasanians unable to mount a cohesive defense.
Caliph Umar ibn al-Khattab directed the main conquest from Medina, appointing field commanders such as Sa'd ibn Abi Waqqas, who won the Battle of al-Qadisiyyah in 636. Khalid ibn al-Walid led the first invasion of Mesopotamia in 633, and Nouman ibn Muqarrin commanded at the Battle of Nahavand in 642.
The conquest led to the decline of Zoroastrianism, which had been Persia's official religion since the Achaemenid Empire around 550 BC. Zoroastrians were made to pay the jizya tax and faced persecution, with reports of scriptures being burnt and priests executed in centers of resistance, prompting many to flee to India where kings took them in as refugees.
The Battle of Nahavand in December 642 was one of the most decisive battles in Islamic history and proved to be the key to Persia. Yazdegerd III had recruited 100,000 veterans and volunteers under Mardan Shah, but the Persians were defeated, after which Umar launched a full-scale invasion of the remaining Sasanian Empire.
Organized Persian resistance ended in 651 when Yazdegerd III, the last Sassanid emperor, was killed near Merv by a local miller for his purse. By then most urban centres in Iranian lands had come under Muslim control, with notable exceptions along the Caspian Sea such as Tabaristan.