There's the Verona of "Romeo and Juliet," and then there's the real Verona in the 16th century.

Many of Shakespeare's plays are set in the Republic of Venice and its cities; apart from The Merchant of Venice and Othello set in the Republic's capital, there is Two Gentlemen of Verona set in the same city as Romeo and Juliet, while The Taming of the Shrew is set in Padua. I think an expert on Shakespeare can give you a better reason for Shakespeare's affinity for Italy; I always wrote it off as Venice's status as the most powerful state in Italy at the time offering the right balance between of exoticism and familiarity, perfect for a play's setting. Venice also had a lax attitude towards protestants, plus the Venetian Ambassador to the English Court, Antonio Foscarini, was a documented theatergoer. All this contributed to cementing Shakespeare's preference for the Veneto instead of other locations in Italy, which was nonetheless the artistic center of Europe and a well-cemented location for fiction at the time. If you'd like to read more, I wrote this answer about the Republic of Venice in the 16h century.

So you might have gathered that Verona was a city belonging to the Republic of Venice. Indeed, Verona was the largest city on the Venetian mainland (however it contests its status as "Second City" of the Republic with Padua, which was smaller but was economically very active and was home to a prestigious university). Needless to say, the real Verona's government was much different than that described by Shakespeare: most cities under Venetian rule were run as miniature versions of Venice's (very limited) republic, and great care was taken to account for different traditions in each city and province. No two cities were run exactly alike. In Verona, the local noble families elected from amongst themselves a fifty-man legislature which sat for a year-long term. From these, a twelve man executive council was selected. Both these bodies had veto power over the Rettore, the supreme executive appointed by the Venetian Council of Ten (the executive branch of the Venetian government) on the advice of the Venetian Senate.

The Rettore's primary responsibilities were overseeing the collection of taxes and what rudimentary public services the state could provide in the Early Modern Era: the Republic paid for free primary schooling, and stipends to Professors of higher education for civil and canon law as well as medicine, who were also exempt from personal taxation (this was the origin of the modern University of Verona, although the most prestigious university in the Republic continued to be that in Padua).

The commander of the local garrison who would also be responsible for public order, called the Capitano, was an army officer appointed from Venice. The post of Capitano di Verona was one of the most important Venetian military posts, especially since after the War of the League of Cambrai about half of the Venetian standing army was be stationed in Verona. Coordination between the province of Verona and its neighbors as well as macro decisions on security (like maintenance of the city's castle and fortifications) were handled from the appropriate offices of the Dominio da Tera in Venice.

Shakespeare's Verona is a very different polity. The play is based on an Italian short story by Matteo Bandello which had been translated into English in 1562. Bandello himself had adapted a story by Luigi da Porto, a Venetian army officer who, while stationed in Udine around 1511, had in turn adapted a short story written by the Southern Italian writer Masuccio Salernitano. Although Masuccio Salernitano had set his story in the Tuscan city of Siena, Luigi da Porto re-worked the narrative to take place in his native Verona.

Luigi da Porto sets his story in 1301, when Verona was the capital of a formidable state in northern Italy ruled by the Della Scala dynasty: governing the cities and provinces of Belluno, Treviso, Vicenza, and Brescia, while Padua was subjected to a form of vassalage, and tight rapports of fealty had been struck with Parma, Lucca, and Cividale. The following century, the Veronese state would be partitioned by the Republic of Venice and the Duchy of Milan, but that's another story.

At the time when Luigi da Porto's story takes place, Bartolomeo della Scala was Lord of Verona. In Shakespeare's Romeo and Juliet, this gets corrupted as "Prince Escales" (probably from the family name's plural form: Scaligeri). The Montecchi family, whom Shakespeare renders as Montague, was an important Veronese noble family which had not always seen eye-to eye with the Scaligeri. Luigi da Porto might have drawn further inspiration from the fact that after being wounded in battle, he finished writing the story from his family estate in the Veronese suburb of Montorso, just a few miles from the Montecchi's ancestral home in the town of Montecchio. Indeed, Luigi's home was in full view of two towers overlooking Montecchio, that today have anachronistically been named, "The Castles of Romeo and Juliet." The Capulets, on the other hand, were probably inspired by a multitude of sources: in the original text their name is rendered as Cappelletti, which Luigi da Porto might have first read about in Dante's Divine Comedy. The Divine Commedy was written immediately prior to the solidification of the Italian political landscape, and in a stanza in the Purgatory the Cappelletti are cited as leaders in the city of Cremona, were are at war with the citizens of Verona led by the Montecchi. Sealing the deal is the fact that Cappelletti could also be the plural form of Dal Cappello, a family of prosperous Veronese spice merchants whose home in Verona, complete with balcony, is known today as "Juliet's House."

I hope I cleared up the status of Verona and why Romeo and Juliet is set there, as well as provided some useful historical context! Feel free to ask any follow-up questions you might have.