A democracy is one where the government is directly chosen by the people (i.e., through elections); a Republic is one where government authority is derived through the will of the people. Thus, a Democratic Republic would be one where ultimate authority and power is derived from the citizens.
United States: The Constitution directly states that the citizens ("We the people") are the foundation of government, and that the government is chosen though democratic elections. Thus, it is a democratic republic.
United Kingdom: is a democracy but not a republic; the government is chosen democratically, but ultimate authority comes from the monarch, not the people.
China: is a republic but not a democracy. Government authority is stated to be based on the will of the people, but there are no democratic elections.