autocorrelationterminologytime series
I'm trying to understand the meaning of cut off and tails off in time series plot of ACF and PACF.
What "Cut off after lag" mean? This about limit?
What "Tails off" mean?
In the example above, the book that I'm using to study, say that is an AR process.
But I can't figure out the meanings of "cuts off" and "tails off"
The ACF and PACF are descriptive statistics showing simple correlation and conditional correlation. They are sometimes useful in identifying ARIMA models that 1) have no Pulses (your series does) and 2) have no deterministic time trends or level shifts ( your series seems to have no trend followed by a period that has trend) and whose parameters and model error variances are constant over time. If you actually post your data in an excel format I will try and help you understand more BUT my initial visual assessment is that this is a time series that may have complications/opportunities that will challenge simple (1960 type !) ARIMA analysis or simple model selection (list-based) schemes that use an AIC/BIC criteria to select/identify a model. We should aspire to keeping things/models simple but not too simple so much so that they are of little value !
EDIT AFTER RECEIPT OF DATA:
I took the 269 monthly values and analyzed it with the automatic option in AUTOBOX. After identifying a global model taking into account various pulse effects a significant difference was found between the first 169 values and the most recent 99 values 
. This is visually obvious.The best model for the most recent 99 values was 
and here 
. The residuals from the model appear well behaved [![enter image description here] . [4]](https://i.stack.imgur.com/Kz01u.png)
The ACF of the residuals suggest a slight possibility of a minor seasonal effect but most likely not important .
. The plot of the actual , fit and forecast using standard ARIMA procedures tp compute uncertainty limits is here 
. Note however from http://www.autobox.com/cms/index.php/blog/entry/you-should-have-50-confidence-in-your-confidence-limits that the forecast limits are flawed on two accounts. Taking into account the possibility of future values being effected by pulses i.e.unusual values and that the estimated parameters are not necessarily the poulation parameters we get a more realistic picture 
You can choose not to provide lags, and let AIC or BIC decide the lags for the ADF test.
ACF/PACF won't tell you much about the number of lags to put in for the ADF test. PACF being cut off after 1 lag indicates that your data is autoregressive order of 1. If PACF is close to 1, then your data probably has unit root, which is what you're going to test with ADF.
ACF is unconditional autocorrelation. Suppose you have a series for the model $y_t=0.9y_{t-1}+\epsilon_t$. Then your ACF will exponentially decay from 0.9 to 0.81, 0.72 and so on. You see positive ACF for 2 lags because $y_t$ and $y_{t-2}$ are correlated through $y_{t-1}$. PACF is partial autocorrelation, which gives you autocorrelation between lags given that other lags are unchanged. So you won't see anything other than the first lag.
Best Answer
"Cuts off" means that it becomes zero abruptly, and "tails off" means that it decays to zero asymptotically (usually exponentially).
In your picture, the PACF "cuts off" after the 2nd lag, while the ACF "tails off" to zero. You probably have something like an AR(2).