confused in practical example of shift invariance?
They talk about the shift-invariance (or variance) of the digital filter. In general, the response of a shift-invariant system is independent of a shift of the input signal. The corresponding output is just shifted by the same amount as the input. So if $y[n]$ is the response to an input $x[n]$, then the response to a shifted input $x[n-n_0]$ is $y[n-n_0]$. If the independent variable is time, then such systems are also called "time-invariant".

Naturally, an adaptive system cannot be shift (time)-invariant, because it changes its characteristics over time, and consequently, it will react differently to the same input applied at different times.

The example in the text refers to the fact that not all problems can be solved by shift-invariant systems only; in certain situations one needs adaptive systems.