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 (holding office with at least two men 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. 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. Perhaps the most significant exception among the forms of today's monarchies is the oligarchical form of election used in the United Kingdom (described under Privy Council).
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 theroetical right to secceed, 2) be economically strong enough to be self sufficient apon seccession, 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 continued to be created even today.
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.