In what sense are "new-Keynesian" models "new" and in what sense are they "Keynesian"? 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"