As you might expect from the name, partial autocorrelation is related to autocorrelation, but there are some subtle differences. Partial means that this is a conditional sort of correlation. In essence, partial autocorrelation measures the correlation of a series with itself at a certain lag after subtracting off any autocorrelations at intermediate lags. You could think of this as the leftover autocorrelation after intermediate correlations have been removed.
The reason that we might want something like this is that we need more than just the ACF to determine the order of our time series model, assuming that it can be modeled by an auto-regressive model. Let's suppose that, using the ACF, we have determined that we ...
