In the immediate aftermath of the most recent Presidential election, President-elect Trump repeatedly crowed about his "historic" win. His utterances suggested that he thought that he had won a broad mandate. His detractors often pointed out that despite winning the election, his main opponent had received more votes. Trumpists, after trotting out a map purporting to show how many counties had received a majority of Trump votes, would respond that "this is a Republic, not a Democracy", as if that explained why we had an electoral college.
I have heard the supposed difference between democracies and republics "explained" to me numerous times over the years, and by "explained" I mean "shouted at me", usually with an admonition to "educate myself". For these people a Democracy is nothing more or less than a Direct Democracy, i.e. "Mob Rule", while a Republic is a government by elected representatives who are guided by a constitution. This is partially true.
"Republic" comes from the Latin Res Publica, "a matter of the people", whereby the government is a public matter and not the private property of a monarch. In other words, in the strictest interpretation, a republic is simply "not a monarchy". The form that a republic can take is quite varied. In the United States, most people who refer to republics are talking about a representative-constitutional government, whereby the executive and legislators are elected by the people for fixed terms and are guided by laws, in our case, a constitution. However who "the people" are can be circumscribed quite broadly. In the early days of the United States only adult, white, male, landowners were permitted to vote. This eventually gave the way to voting rights for all adults, except some felons. In a Socialist, or Communist, Republic, the will of "the people" is assumed to reside in the ruling party, which has elections, but decides ahead of time who may run. In an Islamic Republic, the Quran and the clergy empowered to interpret it decide what's good for the people. Ancient Greek republics limited voting to a citizen class. But even if we accept that the correct or true form of a republic is an elected representative government bound by a constitution, that does not makes democracies and republics mutually exclusive.
"Democracy" literally means power or rule of the people. The simplest form of this is Direct Democracy, often referred to as "Mob Rule", where every decision is voted upon by everyone. Obviously this could only work in very limited situations, such as (very) small towns, or people in small isolated groupings. Direct Democracy almost never exists. A less "pure" form of democracy would involve an elected administrator and/or hired technical people who would keep things running, while important decisions would be voted on. The other end of the democracy continuum would be a system indistinguishable from what we defined above as a representative, constitutional republic: you elect representatives for fixed terms, who are then responsible to run the government according to a constitution. Those who are careful about their words refer to this as a democratic republic, or just a democracy for short. Nobody who calls the United States a democracy really believes that it is a direct democracy or thinks that a direct democracy is a good thing for a large nation.
Nothing about a Democratic Republic, whether you emphasize "democracy" or "republic", guarantees that a system like the Electoral College will exist, or is the ideal system. Various representative democracies/democratic republics use various methods for choosing their head of government, or head of state. Most directly elect their president or prime minister through popular vote. In some countries, notably in the United Kingdom, the Prime Minister is the leader of the party which wins a majority of seats in the House of Commons. We have the system that we do, not because we're a republic, or even strictly because we have a federal system, but because of various compromises arising from competing priorities among the various states. Predominant issues included slavery, small state versus larger states and the tendency for the states to view themselves as sovereign.
Nothing about being a republic guarantees that a tyranny of the majority, the mob rule that detractors of the term democracy so fear, cannot take root. Majorities can still elect representatives that will abuse minorities, and the representatives can be cowed by fear of not being re-elected. What gives the minority rights and protections are the rights and protections that we wrote into our laws early on, such as the Bill of Rights and separation of powers.
Yes, it's important to use our terms accurately, but insisting that our republic is not a democracy is insisting on a distinction without a difference.