The phrase 'time-shift' was used to describe how old-style VHS recorders were used. If Match of the Day was broadcat at ten o'clock, while you were in the pub, you used your video recorder to store the programme, then watched it at midnight.
Of course, similar things happen in real-life: a student who is absent for the end-of-term exam may enjoy a personal time-shift, re-living the examination one week late.
Time Shift in TM4T is explained here.