The advantages of Representative Democracy are for example the fact that while the general population still has an influence on how the country or state is governed, in everyday life they don't need to spend a lot of time on governing, and can focus on other tasks: they're being represented by somebody else.
This leads to a high efficiency of law-making. Direct democracy, where the citizens can vote on laws directly (Switzerland has a form of direct democracy) costs a lot of money, as the referendums are costly to organize, and people need to take time off work at times.
Some advantages are the high risk of corruption and a possibility of discrimination of minorities.