Partial autocorrelation and Autocorrelation difference
Good evening,

Let's begin with a simple example: you have a time series process, for example some process with correlation of up to lag 4.

An order (p) process is time series process where current value depends on previous p values.
You can think for examples of weekly sales in a supermarket, and your AR(4) model in this case shows that your current week sales depend on the previous 4 weeks of sales (on week 1, week 2, week 3, and week 4).

Now, let's get into ACF and PACF.

You have your AR(4) model, which basically tells you that points are correlated up to lag 4, meaning that your correlation between wk 1 and wk 5 is the same as between wk2 and wk 6 (note that the "distance" between them is 4 lags). Similarly, for any number of lags &lt;4 this rule works the same way: corr wk 1 and wk 2 is same as wk 2 and wk 3 and so on. Assume that correlation at lag 1 (wk 1 and wk 2) is stronger than lag 2 (wk1 and wk 3), this means that as number of lags increases, the autocorrelation will also decrease. At each further lag the information from previous lags carries over.
Here is a visualization.

PACF is a completely different concept. What it primarily focuses on is finding out the correlation between two points at a particular lag. In your case, say you want to find the "independent" correlation between wk4 and wk3, this is exactly what PACF will show you.
Here is a visualization.

All together:
ACF data for wk4 will include all the information up to wk3 (wk1, wk2, wk3).
PACF data for wk4 will include "independent" (partial) correlations between wk3 and wk4 only (meaning it will by correlation between these points which wasn't explained by their mutual correlations.

To your point about AR and MA processes, it's the other way around. The lag at which ACF becomes very small is order of MA process. And the lag at which PACF becomes very small is order of AR process.

Hope this helps.