Short answers, pulled from the story.
The Ludus Magnus was the largest gladiatorial training school in Rome, where gladiators from across the empire lived, trained, and prepared to fight at the nearby Colosseum. It also hosted training for venatio, staged hunts and beast fights held alongside single combat.
The emperor Domitian, who ruled Rome 81-96 AD, built the Ludus Magnus in the late first century. He also built three other gladiatorial schools around the same time, the Dacian, the Gallic, and the Matutinus or Morning School, though the Ludus Magnus was the largest of the four.
The Ludus Magnus stood in the valley between the Esquiline and Caelian hills, directly east of the Colosseum. It sat just north of the amphitheater's axis, along the square between the Via Labicana and the Via di S. Giovanni, and was linked to the Colosseum by an underground passage.
The Ludus Magnus fell out of use in late antiquity after gladiatorial combat was outlawed in the fifth century AD. The building was abandoned in the sixth century, and its space was later used to house a small cemetery.
Construction crews near the Colosseum uncovered the buried remains of the Ludus Magnus in 1937. Formal excavations did not take place until 1957-61, and they covered less than half of the original building.
The practice arena at the centre of the Ludus Magnus measured roughly 63 metres long by 42 metres wide, slightly smaller than the Colosseum's own arena. It was surrounded by a small cavea with seating for about 3,000 spectators.