One of the main goals of the Progressive reformers was an increase in democracy. These reformers wanted the people to have more direct control over government in a number of ways. Let us look at some of these.
The Seventeenth Amendment. This amendment provided for the direct election of US senators. Before this was ratified, senators were selected by state legislatures. This amendment provided greater democracy because it gave voters the ability to elect senators directly.
Primary elections. Before primaries were introduced, candidates from the two parties were selected by party activists in caucuses and conventions. Only a few people got any say in the process. Primary elections allowed the party membership in general to vote on who would represent the party. This, too, gave the people more direct control over government.
Initiative, referendum, and recall. These innovations allowed people to vote directly on a number of things. The initiative and referendum allowed them to directly vote on proposed laws. The recall allowed them to decide whether to remove an elected official from office before that person’s term was up. Again, these provided people with more direct control over the government.
In these ways, Progressive reforms allowed for more democracy by giving people more chances to exert direct control over their governments.
The American Progressive movement is generally considered to have originated in the late nineteenth century as a reaction to the abuses of big business and the problems that plagued the lower socio-economic classes of that era.
The first post to this question gave some of the electoral reforms enacted by progressives. The movement as a whole could be considered “democratic” in the sense that it was concerned with helping the “little guys” who could not help themselves. So progressives took on government and big business, often in the form of increased regulation, to help make it more responsive to the needs of people in general. Many people feel that this movement led to Roosevelt's implementation of the New Deal during the Depression.
In modern America we see the influence of Progressivism in movements like Occupy Wall Street, which seek to limit the power and influence of big business.
Some consider Donald Trump to be a progressive in some ways. His call to help American workers by encouraging businesses to stop exporting their jobs abroad is progressive in the sense that it uses governmental power to influence business for the benefit of common Americans. Whether or not this will actually work to America's advantage remains to be seen. Trump, however, is decidedly not progressive in other ways, such as his attitude toward immigration and business regulation.