The problem with the government of the Republic of Venice is that is functioned differently at different times. The Republic was governed according to a complex series of organic ruled and customs, with different institutions supplanting others as the dynamics between power structures changed.

The government at the Republic's peak in the 15th century had most power concentrated in the Consiglio di Dieci, or council of ten, who acted as a sort of executive cabinet. Larger councils of forty people called Quarantia were given specific tasks, such as overseeing monetary policy, or reviewing court cases. Foreign policy and legislation was handled by a Senato, or senate, made up of one hundred and twenty members, many of whom were chosen from existing ambassadors, military commanders, or provincial governors (among the senate, internal commissions called Savi were charged with presenting legislation and setting the senate's order of the day). Election to any of the above bodies was deliberated in the Maggior Consiglio, or Major Council, to which all of-age male members of Venetian families were entitled to sit in (at this time, there were about three hundred families).

Of course, only members of the Major Council could be elected to other councils, meaning that the "Republic" was only a Republic in a very limited sense. You might think that three hundred families all sending their full-grown male members to the Major Council makes for an unwieldy governing system, and indeed it does. However, at any point in time, about a two-thirds of all members of the Major Council would be missing: the Venetian nobility were only small landholders if landholders at all, they still had to partake in commerce for a living. On any given day, most Venetian patricians would be abroad, tending to their trade networks in Constantinople or Alexandria, maybe exploring new ventures in Lisbon or even Bagdad, while others will have been appointed to administrative posts in cities like Candia, Zara or Nicosia overseas, or maybe Belluno, Verona or Rovigo on the mainland (and many did both). Of course, two centuries later, by the time the republic was drawing its last raspy breaths, the decline and ruin of many of these families left them with little to do but go to the Major Council and raise the rabble, and indeed made the efficient running of government impossible.

What about the famous Doge? Elected by a complex system of ballots and electors, his powers were gradually whittled away over the centuries and divided among the various councils. He was the chief magistrate (a sort of one-man supreme court, if you will) and was commander in chief of all the armies in time of war. However as a chief executive his powers were surprisingly limited; he met with the head of the various Quarantia councils to advise them and coordinate their interventions in the senate, but little else. He served as a mostly ceremonial figurehead.

Appointments to executive positions on the Italian mainland, coordination between provinces, and local security decisions were handled from the appropriate offices of the Dominio da Tera in Venice. Another administrative division, called Stato da Mar dealt with Venetian colonies overseas. Although local customs were mostly untouched, Provveditori would be dispatched to administer individual islands such as Negropont, Crete, and Cyprus (and they held much more absolute powers than their counterparts on the mainland), while Castellani would be in charge of the various fortifications (and their garrisons) that dotted the Venetian colonies, insuring safe ports-of-call for her mercantile and military fleets. The central organization of the Stato da Mar, headed by the Provveditore Generale was located on the island of Corfù.

Most cities on the mainland under Venetian rule were run as miniature versions of Venice's (very limited) republic, however great care was taken to account for different traditions in each city and province. No two cities were run exactly alike. In Verona, for example, the local noble families elected from amongst themselves a fifty-man legislature with yearly appointments, and from these a twelve man executive council; these bodies had veto power over the Rettore, the local administrator appointed by the Venetian Council of Ten on the advice of the Venetian Senate.

The commander of the local garrison, called the Capitano, would be responsible for public order. The capitano was an army officer appointed from Venice; the post of Capitano di Verona was one of the most important in the Venetian military posts, especially as after the War of the League of Cambrai about half of the Venetian army would be stationed in Verona. Coordination between the province of Verona and its neighbors as well as local security decisions were handled from the appropriate offices of the Dominio da Tera in Venice.

The above is all covered in the authoritative text on Venetian History: John Julius Norwich's History of Venice. It's a must-read for anyone interested learning about in the City and Republic of Venice.