I've recently been listening to the audio book for 'the storm before the storm by Mike Duncan

I'm sorry.

I also assume it was different depending on different periods in Roman history?

Quite so. Exile in the ancient Mediterranean generally was essentially a voluntary process until rather late. When condemned or accused of a crime one could flee the city, effectively escaping punishment. As the ancient Mediterranean knew no such thing as police forces or public prosecutors the option remained open to a Roman citizen indicted of criminal charges before the people (iudicium populi) or in later quaestio trials to flee voluntarily from the city before sentencing. A more or less mandatory period existed of a couple weeks between condemnation and sentencing that Roman authors speak of as if it were a fundamental right of Roman citizens, even though it was merely custom. No law at Rome prescribed exile as a punishment until Cicero's lex Tullia in 63, which made exile a possible punishment for ambitus. Even after few laws prescribed exile, which was considered a voluntary punishment preferable to the death penalty until Augustus' time. However, since the iudicium populi and especially the quaestiones did not frequently actually use the death penalty once the court process had become regularized exile was effectively the punishment for capital crimes (by citizens at least).

How far a fugitive fleeing the city needed to go was not always entirely clear and changed over time. Originally he seems to have needed only to flee the actual city, although by the actual historical period this seems to have been extended to the Ager Romanus, as well as colonies and towns with full Roman citizenship (it's not clear to me whether this also included the civitates sine suffragia). So exiles often went to Tibur, Praeneste, or the Greek cities in southern Italy. At some point in the first century, possibly already before the Social War, exiles left Italy entirely. Cicero travelled around a bit, mostly in Greece. Milo went to Massilia, where he lived comfortably until he joined then-praetor peregrinus M. Caelius (of the Pro Caelio) in 48 and was killed in battle. Once someone had gone into voluntary exile he was subject to aqua et igni interdictio, in which he was denied all essentials for life by a vote of the people, thereby being prevented from living on Roman soil. He also ceded his citizenship. Some crimes also prescribed that a condemned man, or one who had fled into exile, ceded his property to the state, which confiscated and auctioned it (this is what happened to Milo, and to Cicero). If he fled into exile because of a court indictment (as opposed to say Cicero, who was exiled by a law of the people) he was typically prosecuted, condemned, and sentenced in absentia.

Exile was an upper-class punishment. Very rarely, it seems, were people outside the senatorial or at least equestrian circles even brought to court on capital charges, except in association with high-ranking people whom they had assisted. Additionally, it was fairly onerous to have to deal with arrangements for exile, and the sources repeatedly show that exiles were greatly concerned about their property while they were away. During the Principate this became a regularized, more or less written rule, since honestiores were exiled if condemned on capital charges while humiliores were not. In the Principate the older forms of voluntary exile, while still technically continuing to exist (and occasionally being used), were mostly replaced by deportatio and relegatio. By at the latest the second century AD voluntary flight had become impossible for many crimes, and the condemned were sentenced to exile at remote places like islands. Citizen rights were lost and property confiscated, although enough to live was left the exile, and the exile was transported to the place of banishment, not allowed to go there voluntarily. This was typically permanent (deportatio), though clemency could be given and a temporary banishment (relegatio) existed in the Principate. It's a bit hard to say at what point the old forms of voluntary exile fell out of use. Augustus banished people, both permanently and temporarily, in basically the same way as deportatio and relegatio, but the actual formal procedures are not attested until much later--it doesn't help that exilium describes both the earlier and later procedures.

Besides these there's also "exile" in the form of declaring someone an outlaw, either by proscription or by a declaration of hostis. But that's really rather different...