The Ostrogoths were one of the two major divisions of the Gothic peoples. They migrated into the Empire after the death of Attila and set up a kingdom in Italy under their most famous king, Theodoric the Great.
Like their tribal cousins, the Visigoths, they traced their origins to the ancient Gothic homelands in Scandinavia and northern Poland. The Goths migrated south along the rivers of central Europe in the Second Century and some settled on the steppes north of the Black Sea while others settled in the forests north of the lower Danube. It was the eastern steppe dwellers, or "Greuthungians", who later formed the core of the Ostrogoths.
Under the rule of Ermanaric (350-75 AD) they controlled a wide territory on the Ukrainian steppes and tribes as far away as the Baltic coast paid tribute to them. These steppe-dwelling Goths were farmers and cattle herders, but they adopted many of the customs of their Indo-Iranian nomad neighbours, the Sarmatians and Alans, including their forms of mounted combat.
In the 370s pressure on their territory from the Alans and newly arrived Huns to their east increased and a major defeat led to the (possibly ritual) suicide of the now elderly Ermanaric. Futher defeats by the Huns led the the breakup of Ermanric's former kingdom, with various warbands fleeing westwards. Several entered the Roman Empire during the Visigothic rebellion of 476 AD, with one joining the Visigoths in their victory over Valens at Adrianople in 378 AD. Those who remained were conquered by the Huns and remained under Hunnic domination for almost a century.
The Ostrogoths became a powerful group within the Hunnic kingdom, with Gothic forming a lingua franca among the Huns' various subject peoples and there was a degree of intermarriage between noble Ostrogothic and Hunnic clans. Their king Valamer and his two brothers became key allies of Attila and joined him in his invasions of the Western Roman Empire in 451 and 452 AD. After his death they supported Attila's sons against an alliance of rebel Germanic tribes, but were defeated at the battle of the Nedao River. With the collapse of the Hunnic kingdom the Ostrogoths regained their independence and they entered the Eastern Empire to settle in Pannonia.
Under King Theodoric they fought a series of wars with the Eastern Emperor Zeno, who kept them in check by playing them off against another powerful group of Ostrogoths ruled by Theodoric Strabo. After Strabo died in an accident, Zeno encouraged the Ostrogoths to march on Italy and tackle Odoacer, who had deposed the last Roman Emperor of the West and was ruling Italy as a king. In 493 AD Theodoric defeated and killed Odoacer and ruled Italy in the name of the Eastern Emperor.
Theodoric proved a wise and capable ruler, restoring law and order to Italy and ruling both his Roman and Ostrogothic subjects according to their own laws. He dominated the former Western Empire through a combination of diplomacy, marriage alliances with other tribes and military strength and in his reign learning and culture flourished.
His kingdom fell into disarray after his death in 526 AD however. His daughter Amalaswintha ruled as queen for a while, but was later murdered by her cousin and heir Theodahad. The Eastern Emperor Justinian took this as an opportunity to reconquer Italy and his general Belisarius invaded in 535 AD.
The "Gothic Wars" that followed were long and destructive, raging back and forth during the reigns of successive Ostrogothic warrior kings until the Roman general Narses inflicted the final defeat at Mons Lactarius in 553 AD, where Teja, the last king of the Ostrogoths, died in the fighting.
A few pockets of Ostrogothic resistance held out until 555 AD, leaving a ravaged Italy and weakened Roman army to fall victim to Frankish raids and invasion and conquest by the Lombards.
The Catholic Encyclopaedia on the Ostrogoths
The Ostrogoths: From the Hunnic Domination to the Fall of Ostrogothic Italy
Timeline of the Ostrogothic Kings