What is democracy?

At a basic level, democracy describes a form of government - a way in which people are ruled. What is special about democracy is that, contrary to most forms of government which have existed for most of history, democracies are based on the people ruling themselves rather than kings or just wealthy people ruling the people. Abraham Lincoln describes it best as "government of the people, by the people, for the people"; the people decide on the rules which they themselves will follow.
At a more complex level, it is clear that two types of democracy have existed across history. In ancient Athens, there existed direct democracy, where every single member of the population voted on issues (bar slaves and women - so actually not very many people). In modern democracies, all the people can elect a small number of representatives, but it is these representatives who actually make the decisions on issues. Representative democracy is the only possible form of national-level democracy in modern times, given the size of modern populations making direct democracy impractical.