The Classical school, which is regarded as the first school of economic thought, is associated with the 18th Century Scottish economist Adam Smith, and those British economists that followed, such as Robert Malthus and David Ricardo.
The main idea of the Classical school was that markets work best when they are left alone, and that there is nothing but the smallest role for government. The approach is firmly one of laissez-faire and a strong belief in the efficiency of free markets to generate economic development. Markets should be left to work because the price mechanism acts as a powerful 'invisible hand' to allocate resources to where they are best employed.
In terms of explaining value, the focus of classical thinking was that it was determined mainly by scarcity and costs of production.
In terms of the macro-economy, the Classical economists assumed that the economy would always return to the full-employment level of real output through an automatic self-adjustment mechanism.
It is widely recognised that the Classical period lasted until 1870.
The neo-classical school of economic thought is a wide ranging school of ideas from which modern economic theory evolved. The method is clearly scientific, with assumptions, and hypothesis and attempts to derive general rules or principles about the behaviour of firms and consumers.
For example, neo-classical economics assumes that economic agents are rational in their behaviour, and that consumers look to maximise utility and firms look to maximise profits. The contrasting objectives of maximising utility and profits forms the basis of demand and supply theory. Another important contribution of neo-classical economics was a focus on marginal values, such as marginal cost and marginal utility.
New classical macro-economics dates from the 1970s, and is an attempt to explain macro-economic problems and issues using micro-economic concepts like rational behaviour, and rational expectations. New classical economics is associated with the work of Chicago economist, Robert Lucas.
Keynesian economists broadly follow the main macro-economic ideas of British economist John Maynard Keynes. Keynes is widely regarded as the most important economist of the 20th Century, despite falling out of favour during the 1970s and 1980s following the rise of new classical economics.
In essence, Keynesian economists are skeptical that, if left alone, free markets will inevitably move towards a full employment equilibrium.
The Keynesian approach is interventionist, coming from a belief that the self interest which governs micro-economic behaviour does not always lead to long run macro-economic development or short run macro-economic stability. Keynesian economics is essentially a theory of aggregate demand, and how best best to manipulate it through macro-economic policy.