The Phillips Curve was first hypothesized by William Phillips in 1958. Phillips studied statistical data on the rate of inflation (as measured by the annual change in wages) and unemployment in the British Empire from 1861 to 1957 and discovered that the data resembled a downward sloping curve. This led him to the conclusion that inflation and unemployment are inversely proportional; when one goes down, the other goes up and vice versa.
The main mechanism for this is the short-run stability of wages in an economy.
When inflation rises, the prices of goods and services rise and so companies raise their prices. However, people do not immediately get paid more money because of contracts and the slow speed at which people begin to demand higher wages when prices increase. Since the prices companies can charge go up while the labor costs remain the same, firms pull in a greater profit. This higher profit creates an incentive for firms to hire more workers so they can produce and sell more goods at the higher price. This reduces unemployment. This is the exact relationship predicted by the Phillips Curve.
However, in the long run, the increased hiring reduces the supply of workers and people realize that prices have risen. This allows workers to negotiate higher wages. Now that wages are higher, production costs return to their previous level and the firms receive their original profit. So, in the long run, the Phillips Curve isn't downward sloping: it's vertical. Inflation can move up and down but unemployment will always trend towards the natural rate.
This may all sound a bit abstract but it has practical applications. Despite what many would claim, inflation is actually one of the most powerful forces for economic growth. As a rule, most central banks try to achieve an inflation rate of 1 to 2% to take advantage of the slow adjustment of wages. This actually creates more jobs and greater growth than would normally be obtained by the economy.