Just as with Florence, Venice was a Republic during the Renaissance. Actually, Venice was an empire that controlled land in modern day Italy, a whole lot of sea coast down the Adriatic and countless islands. It enjoyed a stable political climate and thriving trade economy, both of which survived outbreaks of the Black Death and the fall of Constantinople (a major trading partner). Venice was, in fact, so prosperous and healthy that it took someone named Napoleon to undo its empire status...but, that was quite a while after the Renaissance had faded away and had nothing to do with art.
The important part is, Venice (again, like Florence) had the economy to support art and artists, and did so in a big way.
As a major port of trade, Venice was able to find ready markets for whatever decorative arts Venetian craftsmen could produce. The whole Republic was crawling with ceramists, glassworkers, woodworkers, lace makers and sculptors (in addition to painters), all of whom made entirely satisfactory livings.
The state and religious communities of Venice sponsored massive amounts of building and decorating, not to mention public statuary. Many private residences (palaces, really) had to have grand facades on at least two sides since they can be seen from the water as well as land. To the present day, Venice is one of the most beautiful cities on earth because of this building campaign.
Artisan guilds--wood carvers, stone carvers, painters, etc.--helped ensure that artists and craftsmen were properly compensated. When we speak of the Venetian "School" of painting, it's not just a handy descriptive phrase. There were actual schools ("Scuola") and they were highly selective about who could (or couldn't) belong to each. Collectively, they guarded the Venetian art market zealously, to the point that one did not purchase paintings produced outside of the schools. It simply wasn't done.
Venice's geographic location made it less susceptible to outside influences--another factor which contributed to its unique artistic style. Something about the light in Venice, too, made a difference. This was an intangible variable, to be sure, but it had an enormous impact.
For all of these reasons, during the Renaissance Venice gave birth to a distinct school of painting.
The main word here is "light". Four hundred years prior to Impressionism, the Venetian painters were keenly interested in the relationship between light and color. All of their canvases clearly explore this interplay.
Additionally, the Venetian painters had a distinct method of brushwork. It's rather smooth and makes for a velvety surface texture.
It seems, too, that Venice's geographic isolation allowed for a somewhat relaxed attitude toward the subject matter. A great deal of painting dealt with religious themes; there was no getting around that. Certain wealthy Venetian patrons, however, created quite a market for what we refer to as "Venus" scenes.
The Venetian School had a brief fling with Mannerism, but mostly resisted depicting the contorted bodies and torturous emotion Mannerism is known for. Instead, Venetian Mannerism relied on vividly painted light and color to achieve its drama.
Venice, more than any other location, helped make oil paint popular as a medium. The city is, as you know, constructed on a lagoon which makes for a built-in dampness factor. Venetian painters needed something durable! The Venetian School is not known for its frescoes, however.
Well, there were the Bellini and Vivarini families, as mentioned. They got the ball rolling. Andrea Mantegna, though from nearby Padua was an influential member of the Venetian School during the 15th-century.
Giorgione ushered in 16th-century Venetian painting, and is rightly known as its first really big name. He inspired notable followers such as Titian, Tintoretto, Paolo Veronese and Lorenzo Lotto.
Additionally, a lot of famous artists traveled to Venice, thanks to its reputation, and spent time in the workshops there. Antonello da Messina, El Greco and even Albrecht Dürer--to name but a few--all studied in Venice during the 15th and 16th centuries.