I am presenting some ACF and PACF plots to colleagues. I can explain how to interpret the plots and how to determine p and q based on what the plots look like, but I cannot find a simple intuitive explanation of what a PACF plot actually means.
I have read the explanation here, but am finding it a tad long winded: https://people.duke.edu/~rnau/411arim3.htm
An intuitive description of PACF can be "the amount of correlation with each lag that is not accounted for by more recent lags".
Autocorrelation satisfies a property that we could call dampened transitivity. If $x_t$ is correlated with $x_{t-1}$ by some amount $\rho<0$, then $x_{t-1}$ is correlated with $x_{t-2}$ by $\rho$. This implies that $x_t$ is correlated with $x_{t-2}$, although by some amount smaller than $\rho$.
Partial autocorrelation computes the "pure" correlation between $x_t$ and $x_{t-2}$ by removing the "transitive" correlation, that is, the amount of correlation explained by the first lag, and recomputing. For the partial autocorrelation between $x_t$ and $x_{t-3}$, we will remove the correlation with both $x_{t-1}$ and $x_{t-2}$ and recompute, and so on.
You can add some geometric flavour to the explanation. You can picture your time series at each lag as a vector in space. A highly autocorrelated series would look something like this.
The time series with lag 0 could be the vector at the bottom, for instance, the one above the series at lag 1, and the other one is lag 2. The autocorrelation translates to this setting as a large projection of each vector onto each other.
However, what happens if we remove from the original series the projection onto lag 1?
The projection of the remaining length of series 0 onto series 2 is very small. This corresponds to the PACF at lag 2.
