Representative democracy is a form of government founded on the principles of popular sovereignty by the people's representatives. The representatives are charged with the responsibility of acting in the people's interest, but not as their proxy representatives—i.e., not necessarily always according to their wishes, but with enough authority to exercise swift and resolute initiative in the face of changing circumstances. It is often contrasted with direct democracy, where representatives are absent or are limited in power as proxy representatives.
The representatives are chosen by the majority of the voters (as opposed to the majority of the population/eligible voters) in elections. While existing representative democracies hold such elections to choose representatives, in theory other methods, such as sortition (more closely aligned with direct democracy), could be used instead. Also, representatives sometimes hold the power to select other representatives, presidents, or other officers of government (indirect representation).
A representative democracy that also protects liberties is called a liberal democracy. One that does not is an illiberal democracy. There is no necessity that individual liberties are respected in a representative democracy. For example, the Communist states were technically representative democracies who regularly held elections, but did not respect real liberties.