Representative democracy comprises a form of democracy and theory of civics wherein voters choose representatives to act in their interests, but NOT as their proxies - i.e., not necessarily as directed but with enough authority to exercise initiative in the face of changing circumstances. This often involves more powers than a constitutional monarchy or participatory democracy would allocate to the legislators, so almost all constitutions provide for an independent judiciary, and other measures to balance representative power with citizen power:
A representative democracy may provide for recall of elected representatives that voters become dissatisfied with. It may also provide for some deliberative democracy (e.g. Canadian Royal Commission) or direct democracy (e.g. referendum) measures. However, these are not always binding and usually require some legislative action - legal power usually remains firmly with representatives. One halfway is to have an "upper house" that is not directly elected, such as the Canadian Senate, which was in turn modelled on the UK House of Lords.
A European mediaeval tradition of selecting representatives from the various estates (effectively, classes, but not as we know them today) to advise/control monarchs led to relatively wide familiarity with representative systems. Edmund Burke in his speech to the electors of Bristol classically analysed their operation in Britain and the rights and duties of an elected representative.
Representative democracy came into particular general favour in post-industrial revolution nation states where large numbers of subjects or (latterly) citizens evinced interest in politics, but where technology and population figures remained unsuited to direct democracy.
Globally, in 2003, a majority of the world's people live in representative democracies including constitutional monarchy with strong representative branch - the first time in history that this has been true. It has been the most successful form of civics since absolute monarchy. In general, absolute monarchy has become constitutional due to the rise of the power and skill of representatives, sometimes involving political revolutions - but in almost all cases, the representatives come first, and the revolutions have come after.
Normally each representative is elected by, and responsible to, a particular subset of the total electorate: this is called his or her constituency.