A representative democracy in one in which the people elect representatives to make decisions for them, and to stand for their interests when decisions are made. This is different to a direct democracy where people speaks for, and thus represents themselves. One example of a representative democracy would be the British system where constituencies elect MPs to represent them in the House of Commons.
Advantages of a representative democracy are that it makes democracy possible when large numbers of people are involved - it would not be possible to have every person in the UK cast their own vote on every matter. It saves a lot of time, allowing representatives to discuss matters in more depth. Also, not all people understand all issues, and the role of a representative is to take people's opinions and apply those to political decision-making.
There are, however, problems involved. One argument against representative democracy is that representatives are often suspected of working only for their own interests, rather than those of their constituency. Politicians can be seen as elitist, and people's democratic powers alienated. One possible way to solve that problem could be to demand that representatives fully express their own agendas in their campaigns to be elected, though you can never guarantee honesty, especially when dealing in politicians. There could also be more encouragement for people to stand for election, thus giving the electorate a wider choice of representatives.
Apathy is a large problem in representative democracies. When people do not express their views and wishes, a representative cannot know what s/he should take to parliament; they can have no agenda. The prevalence of pressure groups and public demonstrations allows representatives to know what their electorate wants, and if apathy were overtaken by activism and more demonstrations occurred, representatives would be able to stand for those opinions.