What is NAIRU?
Non-Accelerating Inflation Rate of Unemployment (NAIRU) is the unemployment rate consistent with maintaining stable inflation. According to the standard macroeconomic theory, inflation will tend to rise if the unemployment rate falls below the natural rate. Conversely, when the unemployment rate rises above the natural rate, inflation tends to fall. Thus, the natural rate and the NAIRU are often viewed as two names for the same thing, providing an important benchmark for gauging the state of the business cycle, the outlook for future inflation, and the appropriate stance of monetary policy. 
History of NAIRU
During the severe recession of 1974-75 both the inflation rate and the unemployment rate reached some of the highest levels in postwar U.S. history. This experience shook public faith in Keynesianism and played a key role in shaping the subsequent debate about inflation. The warnings of Milton Friedman and other monetarists that attempts to “ride the Phillips curve” might lead to accelerating inflation began to be heeded by more and more people, both inside and outside the ranks of professional economists. The credibility of the monetarist alternative to Keynesian theory was greatly strengthened.
Some Keynesians reacted to the events of 1974-75 by attempting to reinterpret the Phillips curve in a way that reconciled the Keynesian and monetarist views of the inflation-unemployment relation but preserved considerable scope for activist demand management. As early as 1975, Keynesians Franco Modigliani and Lucas Papademos asserted that “the existence of NIRU (the noninflationary rate of unemployment) is implied by both the ‘vertical’ and the ‘non-vertical’ schools of the Phillips curve.” The NIRU was equal to the natural rate. But while monetarists believed that the existence of a natural rate implied that there was no useful trade-off between inflation and unemployment, Modigliani and Papademos interpreted the NIRU as a constraint on the ability of policymakers to exploit a trade-off that remained both available and helpful in the short run. 
Different concepts of NAIRU
The simplest theoretical framework incorporating the NAIRU concept in a transparent fashion is the expectations-augmented Phillips curve, which is also consistent with a variety of alternative structural models. The Phillips curve also has a long empirical tradition of being used as a means of estimating NAIRU indicators. Refinements of the empirical specification led Gordon (1997) to summarize it in terms of the so-called “triangle model” with inflation being determined by three factors: built-in inflation, demand-pull inflation, and cost-push inflation. 
The concept of the NAIRU emerged from the earlier theory of the natural rate of unemployment developed first by Friedman (1968) and others. It views the natural rate of unemployment as an equilibrium position, characterized by labor market clearing (i.e., a position of equilibrium between effective labor supply and labor demand, despite apparent unemployment). The natural rate theory incorporated two ideas that have become very influential. First, the model proposed an inflation barrier – if unemployment fell below the natural rate, high and rising inflation would result, followed ultimately (if the situation was not reversed) by hyperinflation. Second, this inflation barrier was a labor market phenomenon, based on the interaction of the demand for and supply of labor. The demand for labor is negatively related to the real wage, and positively to supply.
The concept of the NAIRU carries over from natural rate theory the ideas that the causes of unemployment lie within the labor market and that the labor market poses the ultimate constraint on the economy’s productive capacity (or, in other word, that the inflation barrier is rooted in the labor market). The phrase itself non-accelerating inflation implies an inflation barrier to full employment. Inflation would quickly accelerate if actual unemployment fell below the inflation barrier posed by the NAIRU. 
- History and Theory of the NAIRU: A Critical Review by Marco A. Espinosa-Vega and Steven Russel – Federal Reserve Bank of Atlanta – Economic Review – Second Quarter 1997
- Estimating the Structural Rate of Unemployment for the OECD Countries by Dave Turner, Laurence Boone, Claude Giorno, Mara Meacci, Dave Rae and Pete Richardson – OECD Economic Studies No. 33 – 2001
- Challenging the Market: The Struggle to Regulate Work and Income – edited by Jim Stanford, Leah F. Vosko – McGill-Queen’s University Press – 2004