In literary terms, apostrophe is when a person (on stage, in the case of a play) addresses a person or other entity (like a god or a city) that is not physically present on the stage.
There are a number of places in Act II of Julius Caesar where apostrophe is used. For example, in Scene 1 of this act, Brutus is talking about why he thinks Caesar is bad for Rome. He addresses Rome as if it were a living entity there on stage with him. He says
Am I entreated(55)
To speak and strike? O Rome, I make thee promise,
If the redress will follow, thou receivest
Thy full petition at the hand of Brutus!
In the part that I have put in bold, Brutus addresses Rome and makes a promise to it. This is a case of apostrophe.
Apostrophe is a literary device in which the writer or speaker addresses someone or something imaginary, either something that is not there or an idea of some kind. Characters in “Julius Caesar” are very fond of it; Brutus just in Act 2 addresses Rome (“O Rome, I make thee promise…”), conspiracy (“O conspiracy, shamest thou to show thy dangerous brow by night?”), and the gods (“O ye gods, render me worthy of this noble wife!”). Some literary critics debate whether that last one is a true apostrophe; when someone addresses the gods, and believes that they are present and can hear him, is that apostrophe or simply prayer? Everyone agrees, though, that it’s not necessary to begin an apostrophe with “O”, but it does sound cool and can’t hurt.