As the Chorus tells us in the Prologue, Romeo and Juliet is set in "fair Verona," an Italian city-state split by a violent feud between its two leading families, the Montagues and the Capulets. In Verona, the Chorus says, "civil blood makes civil hands unclean." The city is ruled by Prince Escalus, a local lord.
The time the play seems to have been the fourteenth century or so, when many Italian polities were in fact divided by the kinds of vendettas portrayed in the story. However, the theme of two lovers from quarrelling families can be found in ancient Greek literature, and was certainly common in Shakespeare's own time.
There is one other city, Mantua, that plays an important role in the story. Mantua is where Romeo lives after he is banished from Verona for killing Tybalt, and an outbreak of plague in that town prevents news of Friar Lawrence's plot from reaching him. Aside from the one scene in Mantua, however, the entire play is set in Verona.