Understanding the HP Filter
The function $\tau$ is the 'line of best fit', the growth trend. The 'residual' $y-\tau$ is the output gap. So, $\tau$ detrends $y$ over time, to give the detrended business cycle $y-\tau$. $\quad y-\tau$ fluctuates around 0&mdash;it is positive when the output gap is positive and the economy is in a boom, and negative when the economy is in recession.

The first term makes the line of best fit follow the points in a more curvy way than just a straight line; the second term makes the line of best fit straighter (the higher $\lambda$ is).