What is the difference between Solow growth, classical growth, and neoclassical growth? My friends and I are taking some introductory macroeconomics courses and these terms keep getting thrown around without being clear how they are related. 

Can someone provide brief definitions for each of these and how they are different?
Classical growth theory: we can put many authors in it but I'd say Ricardo (if we put Malthus aside) is the first one to have describe a real growth theory. For their defenders, growth should somehow end at a certain point in time and is also very unstable for various reasons: Ricardo thinks growth would end because of diminishing returns (cf. Ricardo's On the Principles of Political Economy and Taxation (1817)). In the classical growth theory, you can also include the theory of Harrod-Domar which basically explains that growth is unstable: there is either too much investment in comparaison to savings, or there is not enough investment (cf. Harrod's "An Essay in Dynamic Theory", The Economic Journal (1939)). 
Neoclassical growth: Solow's model is a neoclassical model of growth. It comes in response to the Harrod-Domar model and tries to explain that in the long run, growth is stable (no growth of capital per head).. The only extra growth per head can only come through technological progress (cf. Solosw' "A contribution to the theory of economic growth", Quarterly Journal of Economics (1956)), which is exogeneous.


I encourage you also to look at two other models such as: the AK (accumulated knowledge), which is based on the theory of Solow but tries to make technological changes a endogenous variable in the model (in the Solow model technological change is not the consequence of specific policies); Schumpeterian model created by Aghion and Howitt (1998) which also describes the policies that would be best to stimulate innovation and therefore growth...