My understanding is that a Republic is a nation by which laws apply to everyone. We're doing better at reaching that goal than we used to, but we still have a long way to go.
In truth there are many varying types of republic forms of government, and many varying types of democratic forms of government. The first typically has to do with
who or
what will decide the government of a nation, and the second has to do with
how they decide the government of a nation.
By the definition of the terms, a republic is essentially the opposite of a monarchy:
"a government in which supreme power resides in a body of citizens entitled to vote and is exercised by elected officers and representatives responsible to them and governing according to law."
A "body of citizens", as opposed to a singular head of state like a king or queen (monarchy), a citizen dictatorship, or a military dictatorship. There is also
Stratocracy form of government, which isn't necessarily a military strongman who has taken over rule as a dictator, but a military form of government where the military and state are one and the same.
But "a body of citizens" also doesn't automatically mean a body of ALL citizens. A
Plutocracy (rule by the wealthy class of citizens) is a form of a republican government. An
Aristocracy (rule by the noble class or by nobility) shares similarities with a monarchy, but where there is no single royal head of state, but a class of elite privileged individuals that claims a higher title by birth than the rest of society. There's also the
Meritocracy form of republic government, a system of governance where groups are selected on the basis of people's ability, knowledge in a given area, and contributions to society
All these forms of republican government where privileged few groups of citizens are given rule over the majority, are forms of Oligarchy republican government, and typically tend to not be fair and equitable for all peoples within the governance. But these are ALL republic forms of government.
The United States' form of republic government, is known as In-Direct Democracy, or Representative Democracy. Democracy means "a government by the people - generally meaning the majority of the people - and government in which the supreme power is vested in the people and exercised by them directly or indirectly through a system of representation usually involving periodically held free elections".
Typically democracies tend to come along with a user manual, so to speak - an official document that describes what and how much power the people shall have in their rule, and how that power shall be shared amongst the people. Typically those documents are called "Constitutions".
So the best way to describe the United States' form of government, is as a Republic form of government via Representative Democracy. And that pretty much hits the nail on the head....