What was considered entertainment in the Roman Empire?
This is a very broad question and could take you in various directions. I am going to assume that you are referring to the events that took place in the Roman Coliseum that was the main venue for entertainment provided to the average Roman citizen.
One of the forms of entertainment that took place there was the combat between the gladiators. They were professional fighters who would train for the express purpose of fighting to the death with any opponent with whom they found themselves in the arena. The crowds would watch, thirsty for blood. When two gladiators battled each other to the point that one had the clear advantage, the crowd and ultimately Caesar himself, would signal as to whether the winner should let his opponent live or take his life.
At times, instead of fighting other gladiators, they might face an animal as an opponent.
Circuses were sometimes brought into the Coliseum for the entertainment of the people.
As the persecution of the church increased over the years the Coliseum became a place where Christians would be placed in the arena with wild beasts, or at times gladiators, to be killed as well.