In the 1400s, Florence was a commune run by an elected government composed largely of members of elite families. There was an effective aristocracy, dominated by whichever family had the most influence (i.e. wealth). Families showed off their wealth in various ways, including sponsoring holiday celebrations or public renovations. They exerted their influence through a system of patronage by which they sponsored individuals and guilds in their various business and political endeavors.

The Medici were a banking family with branches all over Europe, even as far as London. Needless to say, their wealth was immense: by some calculations, Cosimo de' Medici (grandfather of Lorenzo the Magnificent) was one of the ten richest individuals who ever lived. This wealth meant that they far outpaced the other families in influence. If you wanted to get a law passed, you needed their blessing; if you wanted to do business, you needed their funding; if you wanted to get elected, you needed them behind your campaign. The Palazzo Medici was the de facto city hall. They never actually held office until the family was granted hereditary Dukedom over the city in the mid 1500s.

I'll wade in here. There are some good answers that perhaps need a bit of tweaking or explanation. I have to disagree with /u/uppityweasel's suggestion that the signorie engaged in informal rule at the time. While many cities in northern Italy were moving towards rule by a single "big man," the way this played out prior to the imposition of grand ducal rule, the character of Medici rule is different from other communes (we can kind of stop calling a good number of these cities communes by the end of the 14th century). While it doesn't receive much attention in English language scholarship (thankfully, this is changing), Bologna can be seen as a leading model for the evolution of the legal means that various Italian cities used to limit the violent factionalism and general repression of the popolo in the 13th and 14th centuries. Balogna was one of the first cities to adopt a signorial regime in 1337 when Taddeo Pepoli seized power (it should be noted that we see the emergence of Signorial regimes as early as the middle of the 13th century as the local "feudal" lords of Arezzo incorporated their powers into the constitutions of the city). Pepoli's power was very much official (something that the Medici did not achieve until the 15th century) and when as Shoa Wray shows in her recent article, "Instruments of Concord: Making Peace and Settling Disputes Through a Notary in the City and Contado of Late Medieval Bologna," his authority was backed up through various legal apparatuses. While many of his arbitrations were extrajudicial, they were backed with the force of law through notaries. Thus, we shouldn't necessarily see the signorie as exercising power through informal means. While this certainly happened, many operated in an official capacity.

A couple of years later, the city of Genoa established a mixed signorial-"republican regime with a heavy guild presence that was established with a sort of quasi-constitution. We can seem the imposition of signorial rule as a gradual evolution of the powers of the offices of podesta and capitano del popolo of the 13th century, all of which was established through legal means. Jones is good for this, but Waley's description is quite a bit more accessible.

As Najemy demonstrates in his excellent history of Florence, the nobles of Florence very much feared the emergence of any sort of signorial regime. Essentially Cosimo laid the ground work for Lorenzo by essentially forcing through constitutional changes that expanded the legislative assembly and packing it with his creatures through his patronage network. Najemy's work also hints at a current trend in scholarship about Florence that questions the extent of the power that the Medici actually wielded prior to Alessandro.

TL:DR, Many cities had a big man, but the Florentine version of this was very different.