Short answers, pulled from the story.
The Arab-Byzantine wars began in the 630s with the first Rashidun raids into Byzantine Syria and continued through the 11th century. The Byzantine recovery under the Macedonian dynasty and the final loss of eastern gains to the Seljuk Turks in 1071 marks the conventional end of the conflict.
Greek fire was an incendiary weapon invented by Kallinikos of Heliopolis, a Christian refugee from Syria. It was used by Constantine IV to decisively defeat the Umayyad navy in the Sea of Marmara during the first siege in 678, and again by Leo III to break the second siege, which ended on the 15th of August 718.
The wars severed Mediterranean trade and drove the Papacy away from Byzantium, which was no longer able to protect Rome while fighting for its own survival. This pushed the Latin church toward the Franks; scholars have argued that the Frankish Empire would probably never have existed without Islam, making Charlemagne an indirect product of Muhammad.
The general 'Amr ibn al-A'as crossed into Egypt at the end of 639 or the beginning of 640 with 3,500-4,000 troops, later reinforced by 12,000 soldiers under Zubayr ibn al-Awwam. Alexandria fell by September 642, extinguishing Byzantine rule in Egypt and depriving the Byzantines of their principal wheat supply.
The Battle of the Yarmuk in 636 was the decisive engagement that opened Syria to Muslim conquest. Muslim forces lured the Byzantines into pitched battle and used the deep valleys and cliffs to turn the fight into a catastrophic defeat, after which Heraclius withdrew from Antioch and Syria passed out of Byzantine control.
The Byzantine resurgence began in earnest at the Battle of Lalakaon in 863, when the general Petronas defeated and killed the Abbasid commander Umar al-Aqta. Under the Macedonian dynasty, the Byzantines reconquered Crete in 961, took Cilicia and northern Syria under Nikephoros II Phokas, and restored Antioch under Basil II, who by 1025 presided over an empire stretching from the Straits of Messina to Edessa beyond the Euphrates.