Hopefully, the title of this question is quite descriptive. Whilst I have a broad understanding of the research agenda of macroeconomics, I don't have a very good picture of how it is divided into various schools and traditions. Is there a way to briefly summarise what exactly New Keynesian macro is and how it relates to the rest of what is going on in macroeconomics?
In essence New-Keynesians adapt micro to macro theory. This is in contrast to new classicals which adapt macro theory to orthodox neoclassical market-clearing microfoundations. New-Keynesians adapt the rational expectations hypothesis but accept that market may fails due to wage and price stickiness and Friedman's natural rate hypothesis. The New-Keynesians, the RBC school and New Classicals focus on issues relating to aggregate supply and have been the dominating schools since the 1970's, especially the new-Keynesians have dominated the last one to two decades. On the other hand "old"-Keynesians and the orthodox monetarists mainly focused on issues relating to aggregate demand and these dominated economic thinking pre-1970's.
As an example of the differences between New-Keynesian New Classicals and the RBC school consider an increase in money: This increase will have real effect in a New-Keynesian model due to these market imperfections. In a new classical model on the other hand money will only have real effects if they are unanticipated (Lucas' Island model) while in a RBC model the increase will only feed into higher prices due to perfect clearing markets and rational expectations.
Note that it has been argued by N. Gregory Mankiw that the New-Keynesian school could just as well have been called New-Monetarist.
For an excellent reference on the different schools, what they represent and their differences see Snowdon and Vane: "Modern Macroeconomics: It's Origins, Development and Current State"
(Old) Keynesian Theory is what you might get taught in undergrad. It comes from behavioral rules and features the good old IS-LM etc diagrams. It rises the Keynesian traps, situations, where the economy might be stuck in downturns of the business cycle for a long time. These situations give rise to, for example, fiscal policy.
Then came the Rational Expectation paradigm, which required a general equilibrium story which is in itself consistent. These guys believed that you can only evaluate the impact of policy if the model is robust to changes in the environment. That is, if you empirically observe a behavior rule (say, consumption), but then the environment changes, the behavioral rule might change. To the extent that we expect policy to adjust the environment, we can only predict policy outcomes if we have a model which is robust to changes in the environment (Lucas critique).
These guys dismissed the old Keynesian Theory because it was built on Behavioral rules, rather than microfundations, and proceeded to create Dynamic Stochastic General Equilibrium models (DSGE), which are supposedly internally consistent and based on micro-foundations: Empirical relationships that are always true and do not change with the environment. This DSGE framework was taken over into most parts of mainstream Economics, most importantly macroeconomics.
New Keynesian Theory is an extension of the simplest possible Real Business Cycle model (which satisfies the DSGE paradigm), which contains most importantly
- market power of firms
- price stickiness
It is designed to give reason for policy intervention, just as the old keynesian theory. However, the intuition and the mechanism are completely different than from the Old Keynesian Story, Cochrane has nice papers and even blog posts on the exact mechanism. It's very debated, not just because its policy implications, but also because price stickiness is again implemented as a "behavioral rule", breaking with the DSGE paradigm.
If you want to know more, you should read up on the papers or ask a more precise question, otherwise this will head nowhere.
The economics and public policy website, Crooked Timber, devoted a question entirely to that subject recently. New Old Keynesianism delineates the differences between
- Old Old Keynesianism,
- Old New Keynesianism,
- New Old Keynesianism and
- New New Keynesianism
such that "New" means “post Global Financial Crisis” and “Old” to refers to a belief that the economy can be in a long-run high unemployment equilibrium.