Short answers, pulled from the story.
Arcadius was the eldest son of Emperor Theodosius I and Aelia Flaccilla. He received the title of Augustus on the 19th of January 383, making him co-ruler for the eastern portion of the Roman Empire.
Eutropius held the position of praepositus sacri cubiculi within the imperial court as a eunuch managing private chambers. In 395, he arranged a marriage between Arcadius and Aelia Eudoxia and later led a successful campaign against Huns in Roman Armenia by 398.
Alaric I commanded Gothic forces during a revolt beginning in 395 and marched toward Constantinople while plundering Macedonia and Thrace. His actions created opportunities for Eutropius and Empress Eudoxia to assume control after Rufinus was assassinated at the Campus Martius parade ground.
John Chrysostom was banished to Abkhazia in the Caucasus by the 20th of June 404 because he criticized extravagant displays of wealth and called Empress Eudoxia Jezebel in a sermon. He remained in exile until his death later that year following condemnation by synods presided over by Theophilus of Alexandria.
Arcadius died on the 1st of May 408 after ruling for twenty-five years. His son Theodosius succeeded him as emperor at a young age, completing the column commemorating victory over Gainas that had stood incomplete since Arcadius's time.