With this notion Alexander Hamilton unfavorably compared pure (or direct) democracy to the republic proposed by the Constitutional Convention in Philadelphia. James Madison defined republic as “a government in which the scheme of representation takes place.” This republic was to be what we call today a representative democracy.
A representative democracy is founded on the principle of elected individuals representing the people. Usually you elect a representative (individual or party) for a fixed term. If you change your mind during the term, you can't do much about it. Also representatives usually stand for a whole package of political objectives. If you don't find your own mix, you need to accept compromises.
On the other hand, a pure (or direct) democracy, where most or all questions are decided by referendum, may be less efficient, is believed to be impracticable on a large scale, and warnings of a mob rule go back as far as Plato. Madison emphasized limitations: “pure democracy, by which I mean a society consisting of a small number of citizens, who assemble and administer the government in person, […].”  Hamilton believed the very character of “ancient democracies, in which the people themselves deliberated, […] was tyranny.”  Representative democracy has always been more than an adequate response to technical limitations in its time; representation is division of labor in politics.
This being said, many people hold up the dream of a pure democracy. New technology such as the internet could place it within reach. Of course, this is only the technical aspect. The remaining question is: Will everybody be able to deal with every question or will people stop participating? Will selfish and superficial decisions predominate? Will the outcome be “tyranny”?
This is where Liquid Democracy offers a promising solution. The basic idea: voters can delegate their vote to a trustee (technically a transitive proxy). The vote can be further delegated to the proxy's proxy thus building a network of trust. All delegations can be done, altered and revoked by topic; e. g. I myself vote in environmental questions, Anne represents me in foreign affairs, Mike represents me in all other areas – but I can change my mind at any time. A dynamic scheme of representation takes place. Anyone can select their own way ranging from pure democracy on the one hand to representative democracy on the other. Basically one participates in what one is interested in but for all other areas gives their vote to somebody acting in their interest. One may make a bad choice once in a while, but can change their mind at any time.
The division of labor – specialization and cooperation – is part of the success story of the human species. Over the centuries division of labor has become increasingly complex and no modern society can exist without. Representative democracy constitutes division of labor in the field of politics. Liquid Democracy makes division of labor available to the voters. While representative democracy remains static, Liquid Democracy offers a dynamic solution. The decision for or against division of labor is left to the individual (i.e. the voter), applies to his or her own vote, can be topic-specific, and can be altered at any time. What about the practical value of this approach? First of all, it provides an alternative organizational concept wherever defined groups (i. e. organizations) decide on issues. For good reasons we will not see any republic being replaced in the foreseeable future (and maybe never will) but apparently Liquid Democracy has the potential to revolutionize decision-making within political parties and thus changing the course of politics.
Political parties can empower their own members, become more attractive to citizens and more responsive to the demand of society. This would be an invitation of a given political party to make politics – in Lincoln's words – of the people, by the people, for the people.