Listing the forms of modern-day democracy is not as easy of a task as one might imagine. We tend to think of democracy and categorize democratic practice through governing institutions. For example, we think of British democracy as a parliamentary form of government, whereas democracy in the United States is a republican form. Different cultures and social structures define democracy in contrasting ways. The Pew Research Institute (May 2019) reports that of the 167 or so countries in the world, more than half exhibit some form of democratic behavior. However, as Pew Research pointed out, these countries have a wide variance in how institutions perform democratically.
Many political scientists now group democracies in four primary categories: Full, flawed, hybrid, and authoritarian. As the types indicate, most nations claim some aspect of democratic practice, even in the most authoritarian states. If one looks only at the full democracies, the class includes direct, representative, constitutional, and monitory. You may not be familiar with monitory democracy. The main feature of monitory democracy is in place of one person, one vote, democratic practice is monitored by groups and interests that guide the governing authorities in making decisions. Monitored democracy is a theoretical iteration of democracy that some political scientists believe special interest groups are more influential in the political process than individuals.
Other, less popular forms of democracy are presidential, religious, social, and authoritarian. Presidential democracy is the notion that a president is the only popularly nationally elected official. As a result, the president has absolute control over government policy. Presidential democracy is what the founders of democracy in America intended to avoid. Religious democracy is when secular laws pass first through the approval of religious authorities. Social democracy is a form of government that combines economic socialism with democratic practices such as elections to parliaments. Authoritarian occurs when elite or wealthy people control the government even when elections happen. The candidates are hand-picked by a small group of people and then voted on by the general elected.
There are several more iterations of democracy. In modern politics, depending on your cultural orientation and country of residence, you will have a very different view of democracy.