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