The Italian City States during the 14th and 15th centuries are frequently depicted as being republics, yet were predominantly controlled by a few powerful families.

I just want to make a small correction here. A "republic" can mean almost any type of government that is not a monarchy, whether hereditary or not. It is a type of government where things are considered publicly-run instead of privately managed as the property of a ruler.

Republican forms of government include both direct democracies (like Ancient Athens or modern Switzerland) and representative democracies (like France, Germany, the USA and UK, South Korea, Japan, Taiwan, etc. although many of these countries have citizens voting directly in referendums from time to time, making them partial direct democracies). Oligarchies are also considered republics, where a small group of people rule over the many (even though they may contain some democratic elements), such as in the ancient city-state of Sparta, the early-modern Polish-Lithuanian Commonwealth, or Apartheid-era South Africa. There are also one-party dictatorships like we see in the People's Republic of China and Singapore (and recently in countries like Turkmenistan). This also includes historical states like Nazi Germany after the Weimar Republic was dissolved with rival political parties officially disbanded.

There are others that defy definition, with blurred lines. For instance, the Republic of ancient Rome was considered by historian Polybius to have a mixed constitution that contained elements of democracy, oligarchy, and even the trappings of Rome's previous monarchy (with executive powers invested in the two elected consuls and the king's former religious authorities and duties granted to the official known as the Pontifex Maximus).

Although it had its own duchies (and of course principalities given the papacy in Rome), Italy during the High Middle Ages, Late Middle Ages, and Renaissance era contained a higher degree of republics than anywhere else in Europe, even though medieval communes stretching from Germany to Spain had elements of democracy (and many of the Imperial Free Cities in the HRE had oligarchies). The origins of constitutional monarchies with parliaments is also an arrangement that stretches back to the High Middle Ages in some parts of Europe (albeit in a nascent stage). Northern Italian republics like Pisa, Venice, Genoa, Florence, etc. had governments that, like ancient Rome, contained elements of democracy, oligarchy, and monarchy wrapped into one. For instance, the power of the doges in the Venetian Republic has led scholars to dub it as a "crowned" republic.

What "Republic" meant to Medieval Italians and what "Republic" means to you or me are very different things. Because of the complex history of Italy in the post-Roman era which precluded strong central government, powerful landed individuals converged onto urban bishops asking for privileges in the marketplace, arbitration of disputes, and in a general sense all the functions which can be expected of government. This rule by voluntary convergence, called the "Commune," would later evolve into the Italian states of the renaissance. However, their republican form was far of a foregone conclusion: by the 16th century there were just as many monarchies in Italy as republics.

Although the communal form of government changed from city to city, all established an executive council of leading families headed by the local Bishop. Initially, there was no fixed system to determine membership; maybe some nebulous "anointment" by the urban bishop had to happen before a dynasty was admitted to the council, but more often than not important individuals could simply leverage social ties in order to make their petitions heard even if they weren't formally members (but in time, this would change). Often the local artisans and merchants guilds were also allowed to participate by sending representatives to the council, or exerted some sort of veto power on things like new taxes. As the system developed, cities came to elect a Podestà, who was often required to be a citizen of a different city, in order to act as a neutral arbiter in council meetings; the mere fact that this could happen attests to the thick web of social relationships across urban communities in Italy, as well as the need for arbitration between the parties that came together in the communal governments. Indeed, the system was far from perfect: I actually discussed the system's shortcomings and instabilities yesterday.

Each city adopted its own laws and covenants: in Milan the landed aristocracy retained a large degree of power, while in Florence the aristocrats were replaced by the upper bourgeoisie. The Venetians, who got a head start in self-rule, adopted a convoluted and continuously evolving system of councils, which eventually became circuitous to the point of ineffectiveness.

So in short, there is no simple, all-encompassing answer. You'll see that each link leads to a different discussion, I highly encourage you to travel down the hyperlink rabbit-hole and feel free to ask any additional questions to clear up any doubts!