Modern republics
A republic is a form of government (and a state so governed) where the head of state is not a monarch. The word is derived from the Latin res publica, or "public affair", and suggests an ownership and control of the state by the population at large. The concept of democracy, however, is not implicit to that of a republic. The republican form of government may involve a limited democracy, where such rights are available only to a limited group of people. In some cases, a republic may be a dictatorial or totalitarian state. The term is also broad enough to include representative democracies.
The use of a republic goes back at least as far as ancient Akkad. The best known ancient republic was the Roman Republic, which lasted from 509 BC until 44 BC. In the Roman Republic, the principles of annuality (holding office for a term of only one year) and collegiality (at least two men held the office at the same time) were usually observed.
In modern times, the head of state of a republic is usually formed by only one person, the president, but there are some exceptions such as Switzerland, which has a seven-member council as its head of state, called the Bundesrat, and San Marino, where the position of head of state is shared by two people.
There is certainly nothing that says that among monarchies and republics one is necessarily more democratic than the other since the powers of the head of state (whether monarch or president) may be purely ceremonial, (although an elected head of state within a democratic system is generally considered more democratic than a monarchy). Monarchs generally reign for life, and when they die they are succeeded by a relative, either chosen by themselves or determined according to set rules. Many modern constitutional monarchies are very democratic, featuring a popularly elected parliament. The presidents of republics, by contrast, are generally elected for a limited term, and their successors are chosen by the body that elected them. These days even non-democratic republics generally claim to be democratic, though the outcome of the election may be assured, and still maintain the ritual of regularly electing their head of state; and frequently in these states, heads of states have left office voluntarily (through resignation or retirement) or been forced out (through constitutional means) by other members of the ruling elite. But there are still some exceptions -- each new Emperor of the Holy Roman Empire, for instance, was elected by the chief princes of the empire, though over the centuries the custom developed of always electing successive members of a particular family to that office.
Republics in the Soviet Union were member states which had to meet three criteria to be named republics, 1) Be on the periphery of the Soviet Union so as to be able to take advantage of their theoretical right to secede, 2) be economically strong enough to be self sufficient upon secession, And 3) Be named after at least one million people of the ethnic group which should make up the majority population of said republic. Republics were originally created by Stalin and continue to be created even today.
States of the United States are required, like the federal government, to be republican in form, with final authority resting with the people. This was required because the states were intended to create and enforce most domestic laws, with the exception of areas deligated to the federal government and prohibited to the states. The founding fathers of the country intended most domestic laws to be handled by the states, although, over time, the federal government has gained more and more influence over domestic law. Requiring the states to be a republic in form was also seen as protecting the citizens' rights and preventing a state from becoming a dictatorship or monarchy.
Republic in classic and renaissance texts
(main article: Classical definition of republic)
Before roughly the 18th century, all known republics were also more or less democratic. This is why in older texts you will often see republic being used interchangeably with democracy. In recent times there have been a large number of not-so-democratic republics, and the definition of the word has become more constrained.
Examples
Currently there is a very large number of republics in the world. A republican form of government can be combined with many different kinds of economy and democracy. Some examples for certain forms of republic are:
Note that while Sparta was considered a republic, monarchy played a part in their government, and while Athens would be considered a republic today, it was described as a democracy in Classical times. In Greek, the distinction between democracy and republic does not exist (since, as noted above, republic is a word of Roman origin). Ancient Greek authors described non-monarchical regimes as oligarchies (if they were ruled by a small group of citizens), tyrannies (if they were ruled by one person who had risen to power by means of coup) or democracies (if right to vote and be elected was given to a large number of citizens, included the lower classes).