The Doge was the elected head of the Venetian state until 1797, when the last one (the 120th) was removed by Napoleon.
Venice was a republic, ruled by an oligarchy with the Doge at its head. Once elected, he reigned until his death. However, his powers were carefully circumscribed, so that he never, for example, had free use of state funds.
The Doge is always recognisable in paintings because of the clothes he wears, in particular the hat with a point at the back, the 'corno ducale'. This fashion seems to date only from the late 14th century.