The concept of "republicanism" should not be confused with the ideologies that are present in the "Republican" party. Although there is some overlap, republicanism is not considered "right-wing" in the same way that the Republican party is.
Republicanism is a political philosophy that stresses the liberty and the inalienable rights of the individual people of a nation. Moreover, republicanism states that the sovereignty of the nation lies with the people. It also rejects aristocracy.
An important difference between republicanism and democracy is that republicanism believes there are some topics that are not subject to vote. While democracy consists on listening to the opinion of the majority, republicanism listens to the majority in matters that do not concern their basic principles.
The United States is more a republic than a democracy. Democracy in the United States is only respected up to a point. Some of the political leaders of the country are appointed, not elected. Moreover, judges (who are not elected representatives) can make decisions that are not popular and are not subject to vote. Democracy in the United States, therefore, is imperfect. On the other hand, republicanism is ingrained in the political system ever since the time of our Founding Fathers, and it is a basic element of the platform of all major political parties.