The first example I can think of is saying 'in the nick of time', as in, barely getting something done before it was too late. For example, 'Dad poured fresh cat food into my empty dish in the nick of time, because if he had waited a moment longer I would have been forced to kill him and do the job myself'.
That sort of thing.
Ok, so why does this 'nick' mean a small amount of time? According to my research, a nick was a unit of measure on a stick that was used to measure time. How this stick was used to measure time is beyond me. I don't bother with measuring time, except when I am waiting on my food or litter boxes to be tended. Anyway, a 'nick' of time sounds stupid, like something a human came up with.
The other use of the word 'nick' is when someone says 'I nicked myself while cutting up your food, Beaker, so I can't feed you this fresh chicken because I am bleeding'. To which I reply 'whatever you've got lots of blood and I don't have much chicken.' According to dictionary.com (human website), the word 'nick' can mean a small groove or cut on the surface. In my mind, the word is derived from a clumsy human who also owned a lot of knives and sharp tools.