There was no ticketing! All events in amphitheatres (including the Colosseum) were paid for by magistrates or wealthy individuals. Access was free for everyone.

Seating was organised by rank. Senators would have the best seats at the front, equestrians would have the seats behind the senators, and the general public would be behind the equestrians.

Suetonius reports that Augustus also made other rules (this is from before the Coloseum, but similar rules would be in effect in later times): "He separated the soldiery from the people. He assigned special seats to the married men of the commons, to boys under age their own section and the adjoining one to their preceptors." (Aug. 44)

Apart from these distinctions based on rank and social status, it was a first come, first serve system. Of course even within a rank there would be difference in social status; that hierarchy would be reflected in the seating, but I'm not aware of any rules for that (other than social convention/pressure).

Some more information on the rank and society is available in P. Garnsey and R. Saller, The Roman Empire: Economy, Society and Culture (Berkeley 1987).