Origins of the Phillips Curve
In 1958, a British economist named A. W. Phillips found a negative relationship between inflation and unemployment. That is, years of high inflation are associated with low unemployment. This negative relationship has been found for other countries and is known as the Phillips curve. The Phillips curve appears to offer policymakers a menu of inflation and unemployment choices. In order to achieve lower unemployment, one needs only choose a higher rate of inflation.
What is the Phillips Curve?
The Phillips curve shows the combinations of inflation and unemployment that arise in the short run as shifts in the aggregate-demand curve move along a short-run aggregate-supply curve. For example, an increase in aggregate demand moves the economy along a short-run aggregate-supply curve to a higher price level, a higher level of output, and a lower level of unemployment. 
Although it was discovered to be invalid in the long run, Phillips curve has been used in varied and modified forms to understand inflation. The evolution of Phillips curve doctrine before 1975 is widely accepted and no longer elicits much debate. The discovery by Phillips and his disciples Samuelson‐Solow of an inverse relationship between inflation and unemployment briefly suggested an exploitable policy tradeoff that was destroyed by the Friedman‐Phelps natural rate hypothesis of the late 1960s.
The development of the Phillips Curve
However, after 1975 the Phillips curve literature bifurcated into two lines of research which since then have not had any connection between them. Along the left fork in the road (more commonly known as Gordon’s triangle model), it was revived by importing micro demand and supply analysis into macroeconomics. There was no assumption that unemployment and inflation are negatively correlated.
The right fork in the road built models responsive to current and anticipated changes in policy. Important elements of this side include policy credibility, models of the strategies of policymakers and private agents forming expectations, and the New Keynesian Phillips Curve (NKPC) which derives a forward looking Phillips curve from alternative theories of price stickiness. The common feature of these theories is the absence of inertia, the exclusion of any explicit supply shock variables, the ability of expected inflation to jump in response to new information, and alternative barriers to accurate expectation formation due to such frictions as rational inattention. 
More recent economic analysis of the trade-off between inflation and unemployment considerably modifies the simplistic stable Phillips curve theory of the past. It is now recognized that both labor and management incorporate their expectations about the likely future rate of inflation into their bargaining positions in wage negotiations, since it is the real purchasing power of the future wage payments rather than simply the nominal number of currency units that matter to them. As average inflation rates in the industrial countries rose over time, labor and management eventually began adjusting their inflationary expectations for the future upward as well, and wage settlements began to reflect these higher long-term expectations.
As a result, the Phillips curve shifts upward and outward to a higher level, further away from the origin of the graph. Government policy makers now can lower unemployment rates (in the short to medium term) only by creating a rate of inflation even more rapid than the public previously had come to expect — and this works for only so long as it takes the public to adjust their expectations upward in the light of their new experiences. 
- Principles of Economics by N. Gregory Mankiw – South-Western Cengage Learning – December 2010
- The History of the Phillips Curve: Consensus and Bifurcation by Robert J. Gordon – Northwestern University, NBER, and CEPR – March 7, 2009