The terms "left" and "right" appeared during the French Revolution of 1789 when members of the National Assembly divided into supporters of the king to the president's right and supporters of the revolution to his left. Generally, the left-wing is characterized by an emphasis on 'ideas such as freedom, equality, fraternity, rights, progress, reform and internationalism', while the right-wing is characterized by an emphasis on 'notions such as authority, hierarchy, order, duty, tradition, reaction and nationalism'
On a social level, the right generally believes that a hierarchy in society is natural and normal. For them, the idea of a class system whereby the richer employ the poorer in a triangle is healthy because everybody benefits and it has historically been the natural order of things. Their economic policy (generally speaking) complements this, by promoting a form of capitalism which gives more economic freedom to big businesses. This means fewer regulations and lower taxes. They argue that as big businesses grow they create jobs for the working class and gradually everyone gets richer.
The left on the other hand is more keen to reduce the gaps in society between the rich and the poor. They believe society should work together to help everybody, with schemes like single payer health care and education. They argue the rich should pay proportionally more for this through taxes, thus reducing the gap between rich and poor. They also prefer to regulate 'the Market', which means distributing the wealth created on the stock market and in banks so that it does not all end up in the pockets of millionaires but some of it goes to solving the country's problems.
The easiest way to think about it is the idea of 'State vs. Market'. The state is the government, using its power to solve problems that only the government can; healthcare, policing, schools etc. Those departments are the 'public sector'. The market is the capitalist force which creates wealth, such as business transactions and banking. This is the 'private sector'.
I hope this helps.