A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application.
The opposite is true for time invariant systems (TIV).
There are many well developed techniques for dealing with the response of linear time invariant systems, such as Laplace and Fourier transforms.
However, these techniques are not strictly valid for time-varying systems.
An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change.