In a republic
there is representative democracy, so that the leaders are chosen by the people and the laws are made by the elected leaders. However, the individual citizen has fundamental, natural rights which the government may not violate, AND THE PEOPLE CANNOT VOTE TO VIOLATE. These rights exist inherently in being a human. In this sense there is some absolute limit on what laws can be enacted, and what actions may be taken against any individual.
In a straight democracy, it's whatever the majority votes. They could vote to execute all red-headed people, and it would be done--nothing the redheads could do about it. What in theory makes America a republic, then, is the existence of inviolable civil rights belonging to individuals and not dependent in any way on the decisions of government or the votes of the people.
Your anxiety, which is reasonable, appears to arise from the fact that it is becoming allowable to abridge some individual rights under pressure from officials and voters. To that extent we are less a republic than we were or ought to be.