Stationary autoregressive (AR) processes have theoretical autocorrelation functions (ACFs) that decay toward zero, instead of cutting off to zero. The autocorrelation coefficients might alternate in sign frequently, or show a wave-like pattern, but in all cases, they tail off toward zero. By contrast, AR processes with order p have theoretical partial autocorrelation functions (PACF) that cut off to zero after lag p. (The lag length of the final PACF spike equals the AR order of the process, p.)
Moving average (MA) process
The theoretical ACFs of MA (moving average) processes with order q cut off to zero after lag q, the MA order of the process. However, their theoretical PACFs decay toward zero. (The lag length of the final ACF spike equals the MA order of the process, q.)
Stationary mixed (ARMA) process
Stationary mixed (ARMA) processes show a mixture of AR and MA characteristics. Both the theoretical ACF and the PACF tail off toward zero.