ACF describes the autocorrelation between an observation and another observation at a prior time step that includes direct and indirect dependence information.
$\bar{y}$ is a mean of time series $y$
The autocovariance function at lag k, for k ≥ 0, of the time series is defined by:
$s(k) = \dfrac{1}{n} \sum_{i=k+1}^{n}(y_i - \bar{y})(y_{i-k} - \bar{y})$
The autocorrelation function (ACF) at lag k, for k ≥ 0, of the time series is defined by:
$r(k) = \dfrac{s(k)}{s(0)}$
$s(0)$ can be interpreted as the variance ($s(0) = \sigma_{y}^2$)
$r(0)$ is always 1 and should be ignored
Consider a time series that was generated by an autoregression (AR) process with a lag of k.
We know that the ACF describes the autocorrelation between an observation and another observation at a prior time step that includes direct and indirect dependence information.
This means we would expect the ACF for the AR(k) time series to be strong to a lag of k and the inertia of that relationship would carry on to subsequent lag values, trailing off at some point as the effect was weakened.
Lag is the difference between measurements. If we have daily data, the lag value k=2 means, that the autocovariance function is the normalized sum of the covariance between 9.07 and 07.07, 08.07 and 06.07, 07.07 and 05.07, .$.$. 01.01 and 30.12