If
$ x(t) \to System \to y(t) $
implies,
$ x(t-t_0) \to System \to y(t-t_0) $
for any $ x(t) $, then the system is time invariant.
Example of a time-invariant system:
$ System = \sqrt{t} $
$ x(t) = t + 2 $, $ t_0 = 1 $
$ x(t) \Longrightarrow System \Longrightarrow \sqrt{t + 2} $
$ x(t-t_0) \Longrightarrow System \Longrightarrow \sqrt{t + 1} = \sqrt{t +2 -t_0} \therefore $ the system is time-invariant.
Example of a time-variant system:
$ System = t^2 $
$ x(t) = t + 2 $, $ t_0 = 1 $
$ x(t) \Longrightarrow System \Longrightarrow (t + 2)^2 = t^2 + 4t + 4 $
$ x(t-t_0) \Longrightarrow System \Longrightarrow (t + 1)^2 = t^2 + 2t + 1 \neq t^2 + 4t +4 - t_0 \therefore $ the system is not time-invariant.
