The curve that illustrates the trade-off between inflation and unemployment is the Phillips curve.
Economist A. W. (Bill) Phillips identified an inverse relationship between unemployment and the rate at which money wages changed in Britain. His landmark paper, published in 1958, examined British data from 1861 to 1957. The idea was later expressed as a relationship between inflation and unemployment by Paul Samuelson and Robert Solow in 1960.
The basic short-run interpretation is that stronger demand can reduce unemployment while increasing wage and price inflation. However, the relationship is not a permanent menu from which policymakers can freely choose. Milton Friedman and Edmund Phelps argued in the late 1960s that expectations of inflation alter the trade-off, and that unemployment tends toward a longer-run natural rate or NAIRU.
A common mix-up is confusing the Phillips curve with the Laffer curve. The Phillips curve concerns inflation and unemployment; the Laffer curve concerns tax rates and tax revenue. Modern macroeconomic models still use versions such as the expectations-augmented and New Keynesian Phillips curves, but the simple permanently downward-sloping version is considered inadequate, especially after the stagflation of the 1970s.