Some may say that's an oxymoron, but it doesn't invalidate that the US currently falls into that general category. There are in fact countries with the name in their titles, such as the Democratic Republic of the Congo, but the term is more broad. Think of it more like a genre of governments, of which Constitutional Republics is one species.
You can also be a Constitutional Monarchy, like Great Britain. However, even this form of government ultimately can be classified as Democratic Republic because power lies with the Houses of Commons and Lords. Great Britain runs it's government through elected officials, making it a de facto Democratic Republic.
You could also include Israel, which has no Constitution per say, but maintains documents similar to the United State's Bill of Rights called the Basic Laws of Israel.
All citizens of Israel - regardless of their race, religion or sex - enjoy equal human rights and protection under the law. This principle dates to the founding document of modern Israel, the May 1948 Declaration of the Establishment of the State of Israel. This declaration of independence proclaimed that the State of Israel will be based on freedom, justice and peace as envisaged by the prophets of Israel; it will ensure complete equality of social and political and human rights to all its inhabitants irrespective of religion, race or sex; it will guarantee freedom of religion, conscience, language, education and culture; it will safeguard the Holy Places of all religions; and it will be faithful to the principles of the Charter of the United Nations.
The Declaration maintains its influence as a guiding principle in the interpretation of laws. Its centrality was acknowledged in the1992 Basic Law: Human Dignity and Liberty,(link to next section) which explicitly provides that the human rights included in the law shall be interpreted "in the spirit of the principles in the Declaration of the Establishment of the State of Israel."Soon after the founding of the State, the Knesset began to enact a series of basic laws relating to all aspects of life. As well as the basic laws outlining the primary features of government, additional laws have been passed which deal with fundamental rights, such as the Basic Law: Human Dignity and Liberty.
Israel uses a parliamentary system as well as courts, like Great Britain, to govern. It's head of state is it's Prime Minister, head of the majority coalition that forms after elections. In that way, it too becomes a Democratic Republic.
So I use the label widely, in particular, because democracy solely means that each person has a vote.
Often times democracy itself can be corrupted or tyrannical. You can have the tyranny of the majority over the minority, and even visa-versa.
Or you can have a Dictator, such as Saddam Hussein, pretend to use Democracy to remain in power.
Or you can have a Dictator abuse the Democratic Republic, and turn it into a Democratic Autocracy- as spoofed in the movie, The Dictator, by Sasha Ben Cohen.