[This is an edited version of something I've posted here in the past. I'm hoping others will be motivated to add to (or correct) this history.]
The term "New Classical economics" is often used as though it is one of the dominant models in macroeconomics, but the term has a very specific meaning and refers to a class of models that is no longer popular.
The New Classical model has four important elements, the assumption of rational expectations, the assumption of the natural rate hypothesis, the assumption of continuous market clearing, and an assumption that agents have imperfect information (imperfect information drives cycles in these models). The imperfect information assumption was quite clever in that it allowed proponents of this model to explain correlations between money and income without acknowledging that systematic, predictable monetary or fiscal policy would have any effect at all on real output and employment (put another way, only unexpected changes in monetary policy matter, expected changes are fully neutralized by private sector responses to the policy).
The New Classical model is important for the foundation it provided for later models, the movement in macroeconomics toward microeconomic foundations and the use of rational agents within macro models in particular, but the model itself could not simultaneously explain both the duration and magnitude of actual cycles. It also had difficulty explaining some key correlations among macroeconomic variables, and it was difficult to understand why a market for the absent information did not develop if the consequences of imperfect information were as large as the New Classical model implied. If you are at Chicago where these models were popularized, markets pop up as needed and the fact that there was no market to help agents avoid the confusion that drives the New Classical model was a strike against them. In addition, one of the model's key results that only unexpected changes in money can affect real variables did not hold up when taken to the data (though there are still a few die-hards on this). So the profession moved on.
The New Classical model had replaced the old Keynesian model after the old Keynesian models' shortcomings were blamed, at least in part, for the problems we had in the 1970s. The model was also abandoned for theoretical reasons that will be described in a moment.
But while the New Classical economists were having their day in the sun, the Keynesians were quietly working behind the scenes to fix the problems that caused the old Keynesian model to go out of favor (or not so quietly in a few cases). The old Keynesian model had a poor model of expectations. If expectations were considered at all, they were usually modeled as a naive adaptive process. In addition, it was not clear that the assumptions and relationships embedded within the old Keynesian model were consistent with optimizing behavior on behalf of households and firms. The New Keynesian model solved this by deriving macroeconomic relationships from microeconomic optimizing behavior, and by adopting the rational expectations framework. And the New Keynesians made one other important change. In order for systematic monetary policy (e.g. following a Taylor rule) to affect real variables such as output and employment, there must be some type of friction that prevents the economy from immediately moving to it long run equilibrium value. The friction in the New Classical model is informational, agents optimize given the information that they have, but because the information is imperfect the decisions they make take the economy away from its optimal long-run path. In the New Keynesian model the friction that gives monetary policy its power is sluggish movement of prices and wages (generally modeled through something called the Calvo pricing rule). This friction is somewhat controversial and the precise degree of price rigidity in the economy is the subject of intense research.
Many people who use the term New Classical -- a natural counterpart to the term New Keynesian -- seem to have in mind some version of a Real Business Cycle model where prices are, in fact, assumed to be fully flexible, agents are rational, all markets clear, policy is neutral, etc. In these models, actual output is always equal to potential (so there's no need for policy to do anything but maximize the growth of potential output, hence the supply-side orientation of advocates of this approach). Potential output moves over time in response to productivity and taste shocks, i.e. supply shocks, and that is the source of business cycles in thsi class of models. Demand shocks, which drive business cycles in New Keynesian models (as well as New Classical and Old Keynesian models) have little or no effect on real output and employment.
I was recently labeled as a "neoclassical" economist, so let me end by making it clear that not all of us believe that assuming fully flexible prices and continuous market clearing is the proper way to model the economy. Prior to the crisis I was an advocate of sticky price/sticky wage New Keynesian models, and quite resistant to pure Real Business Cycle approaches. But I am less of a fan of the New Keynesian model than I once was. I still think it's a good model to explain mild fluctuations of the type we had during the Great Moderation, and I still think the tools and techniques macroeconomists use, what is collectively labeled DSGE models, are the right way to go (though I would still like to see competing models challenge this view). But to be useful in a crisis like we just had the models have to be amended to better connect the real and financial sectors -- the connection between breakdowns in financial intermediation and the real economy needs to be improved -- and people are working hard to try to solve problems. Will they succeed? I certainly hope so.