In thinking about this question, we must be sure to distinguish between republicanism with a small “r” and Republicanism with a capital “R.” This is a question about republicanism with a small “r,” not about the Republican Party, which did not come to exist until the 1850s.
Republicanism is essentially the form of government that we have always had in this country. Republicanism is very similar to democracy, and most people today would not make any distinction between the two. Republicanism believes that the people should be sovereign. They should have control over their government. They should be the ones who are ultimately in control because they are the ones who vote and participate in politics in other ways.
In addition, republicanism holds that the government should have only limited powers. Republicanism believes in a government that is prevented from doing certain things, regardless of what the majority of the people want. Therefore, a republican form of government has things like civil liberties that are protected from the government.
We can see that America is still very much governed by republican principles. We believe in the power of the people and in civil liberties as much as (or even more than) we did in the time right after the Revolution.