A democratic republic is a form of government which operate on principles adopted from a republic and a democracy. It is not a cross between two entirely separate systems, rather democratic republics function on principles shared by both republics and democracies. For example both the US and India are democratic republic countries with two different styles of democracy.
The president of United States is elected by the people to be the most powerful executive of the country under the presidential democracy after every 4 years. In case of India, the President is the nominal head of the state indirectly elected by the elected representatives of the people periodically after every five years.
India follows the parliamentary democracy in which the Prime Minister is the leader of the majority party in the lower house of the parliament. A democratic republic country has not only the elected head of the state changing periodically but also that elections are democratically elected by involving the common citizens of the country casting votes. From that yardstick, China is the republic country but not a democratic country because citizens of China do not participate in any general elections. President of China always comes from the communist party as it is called one party dictatorship.
There are many countries which use the word ‘democratic-republic’ for their country but are not so. Many countries that use the term “democratic republic” in their official names (such as Algeria, Congo-Kinshasa, Ethiopia, North Korea and Laos considered as hybrid regimes instead of being pure democratic-republic countries.
“People’s Republic” is a title used by those sovereign states which have formed socialist governments under the influence of Marxist-Leninist ideology. Most of them are ruled under strict dictatorship without giving any opportunity to their people to participate in free elections.
They love to call themselves socialist states in their constitution and are guided by theory of populism. Of course, their head of the state is periodically changed and are not hereditary in succession but they are not broadly elected by the people as it happens in the democratic regimes. Some of the current officially socialist states that include the words people’s republic in their full names are: People’s Republic of China, Republic of Cuba and Socialist Republic of Vietnam.