|Economists never get together at conventions to standardize naming conventions, but in the vernacular there's sort of a family relationship between the "neoclassical synthesis," "neo Keynesianism," and "new Keynesianism."
the neoclassical synthesis arose after WW2, with Paul Samuelson: the idea was that the government and the central bank would maintain rought full employment, so that neoclassical notions -- based on scarcity -- would apply.
neo-Keynesianism (often contrasted with the post-Keynesianism of Paul Davidson, et al) is based on the synthesis but puts more emphasis on microfoundations, the use of Walrasian general equilibrium theory in macroeconomics. This developed over time.
new Keynesianism (associated with Greg Mankiw, now Bush's CEA chair) is a response to the Robert Lucas/new classical school, which criticized the inconsistencies of the neo-Keynesian school in light of the concept of "rational expectations." The new classicals combined a unique equilibrium (at ful employment, natch) with rational expectations. The new Keynesians say: we have all sorts of "microfoundations" that indicate that markets don't clear because of price stickiness, so there's no unique equilibrium in the short run, so the rational expectations-based critique doesn't apply.