Regarding views on a democracy versus a republic

So, I hear many libertarians bemoan the fact that "we used to be a republic, now we're a democracy." I don't have an opinion on this one way or the other because I have no idea what it's supposed to signify. Are people with such views pining for the days when direct election of senators or the president didn't exist? I guess I just don't know what the substance of such an argument is and I'd be very appreciative if anyone could explain its merits and/or disadvantages to me.