A liberal democracy (that is, one that champions the development and well-being of the individual) is organised in such a way as to define and limit power so as to promote legitimate government within a framework of justice and freedom.

According to Jim Kilcullen:

In Ancient Greece some cities were democracies, others were oligarchies. Democracy meant rule by the people, oligarchy meant rule by the few. So a city was a democracy if: city affairs were subject to an Assembly; to which all male citizens belonged; and in which decisions were made by simple majority vote.

According to Dr John Hirst:

A democracy is a society in which the citizens are sovereign and control the government

According to Joseph Schumpeter:

The democratic method is that institutional arrangement for arriving at political decisions in which individuals acquire the power to decide by means of a competitive struggle for the people’s vote.

Different Types of Democracy:

Direct democracy
Representative democracy
Constitutional democracy
Monitory democracy

The key difference between a democracy and a republic lies in the limits placed on government by the law, which has implications for minority rights. Both forms of government tend to use a representational system — i.e., citizens vote to elect politicians to represent their interests and form the government. In a republic, a constitution or charter of rights protects certain inalienable rights that cannot be taken away by the government, even if it has been elected by a majority of voters.

A constitutional republic is a state where the chief executive and representatives are elected, and the rules are set down in a written constitution.

Constitutional monarchies are a special case: even though the monarch is not elected, the people still elect other governing bodies. The constitution also limits the power of the monarch.

Republics and democracies both provide a political system in which citizens are represented by elected officials who are sworn to protect their interests.

According to US Founding Father, James Madison:

“It [the difference] is that in a democracy, the people meet and exercise the government in person: in a republic, they assemble and administer it by their representatives and agents. A democracy, consequently, must be confined to a small spot. A republic may be extended over a large region.”

Most modern countries these days run a mixture of multiple systems, and employ elements from both democracy and republic. Representative democracy is closely associated with a republic.

Alexander Hamilton’s letter of May 19, 1777, to Gouverneur Morris.

“But a representative democracy, where the right of election is well secured and regulated & the exercise of the legislative, executive and judiciary authorities, is vested in select persons, chosen really and not nominally by the people, will in my opinion be most likely to be happy, regular and durable.”

Most republics, including USA, function as blended “representational democracies” featuring a democracy’s political powers of the majority tempered by a republic’s system of checks and balances enforced by a constitution that protects the minority from the majority.

(This information is taken from various sources, to piece together constructive knowledge about political systems widely used in the West)