Modern politics and labels aside, a Republic is basically a representative democracy as opposed to a pure democracy where every citizen voted on every little thing.
Our founders had a real fear of democracies, and "mob rule" and that, along with the Hamiltonian/New England v. Jeffersonian/Virginian, stuff all played into a lot of discussions including the "compromise" where the less populous south got to count non voting slaves as 3/5 of a person.
My guess is this point came up talking about the popular vote and the electoral college, which was a choice to elect leaders by representatives rather than "democratically", I.e., by popular vote.
There are more registered democrats in fewer but bigger states than there are Republicans in smaller states, so you might hear democrats calling for a popular vote with Republicans saying that the founders never intended a democracy, but a Republic.
Republicans are not wrong on this one, IMO, but we've been arguing about how "United" the States should be since before the Constitution was enacted, and how voting should work, and we've tweaked it before, both by Amendment and by practice.
I have never heard a mainstream Republican actually talking about disenfranchisement in a Republic, but by picking electors by state as well as by population, and then letting the States decide how theyre going to pick their electors and cast their votes,we are making some individual votes "count more" than other votes and that is less "democratic."
Hope that helps. These days, it's really hard to figure out what people are talking about, and a lot of the time, they're repeating words, phrases and arguments they've heard on cable TV without even understanding what they're saying, being able to apply it in any given context.
Neither Hannity nor Maddow is a good substitute for a civics class, IMO.
