Autocorrelation and partial autocorrelation are a type of measures of association between current time series and past time series values. Both of these provide an indication that older time series values are more useful in predicting future values.

Autocorrelation is the correlation of a Time Series with lags of itself. This is a significant metric because:

1. It is able to demonstrate the difference between a past state of time series observations and current state of time series observations and how much of lag impact. In below diagram time series of “Air Passengers” dataset is considered and ACF plot is drawn. As ACF lines are consistently outside of the dotted line, it indicates a significant correlation of lags in the series.
2. It also provides the aspect that the series is stationary or not. Autocorrelation plot is also called correlogram. If there is a stationary element, then correlogram will fall to zero, else if there is no stationary element, then correlogram will fall gradually slowly.  

While comparing current time series steps to that of prior time series steps, there can be direct and indirect correlations. The indirect correlations are a linear function of correlation of the observation. There could be INTERVENING time series steps. PACF or Partial autocorrelation tries to remove the EFFECT of correlation due to shorter lags.  

Both ACF and PACF are useful while trying to understand which model approach could be a RELEVANT and better fit for a prediction solution.