Short answers, pulled from the story.
The Gothic War was triggered by the murder of Amalasuntha, regent of the Ostrogothic Kingdom of Italy, in early 535 at the hands of her cousin Theodahad. Her death gave Emperor Justinian I a formal justification for war, though the deeper cause was his ambition to recover the former western provinces of the Roman Empire.
Belisarius was appointed commander in chief for the Italian expedition with 7,500 men. He led the first phase of the war from 535 to 540, capturing Sicily, Rome, and ultimately Ravenna. The general Narses commanded the decisive second phase beginning in mid-551.
Totila, proclaimed Gothic king in the early autumn of 541, exploited the Plague of Justinian, ongoing Byzantine-Persian conflicts, and the incompetence of Byzantine commanders in Italy. He destroyed Byzantine armies at the battles of Faventia and Mucellium, captured Naples in 543 on favourable terms, and entered Rome on the 17th of December 546 after his men opened the Porta Asinaria.
Narses defeated and killed Totila at the Battle of Taginae, fought near the village of Taginae after Narses led a force of 20,000-25,000 men into Italy in 552. The last Gothic resistance at Campsa, near Naples, capitulated in the spring of 555, and the final holdouts at Verona and Brixia fell in 562.
The war devastated Italy's cities and population while draining Byzantine resources needed to defend the Balkans and western Asia. Just three years after Justinian's death in 565, the Lombards invaded and seized most of mainland Italy. Dalmatia and much of Greece were later lost to the Slavs and Avars, suggesting the Italian campaign came at a severe strategic cost.
Italy fell into a prolonged period of depopulation and urban decline. The Lombards invaded in 568 and took most of the peninsula. Byzantine control shrank to the Exarchate of Ravenna and parts of southern Italy. Rome remained under Byzantine control until 751. Byzantine influence in coastal southern Italy persisted until the late 11th century, while the interior was ruled by Lombard dukes at Benevento, Salerno, and Capua.