I have no idea of the origins of this word, so my understanding of it may be wrong, but where I come from its a phrase that means you're so completely tired that you're over tired or totally beat. I can only hguess its a reference to the fact that a lot of dogs seem to sleep a lot and thus appear to always be tired and in need of sleep.
I pulled a double shift last night and when I got home the neighbours were having a rowdy party with loud music and I couldn't get any decent sleep. I am already dog tired and I have another double shift tomorrow.
It is said of the situation where a person has the bad luck to make contact with his testicles against an undefined surface or object, intentioned or not.
Given the nature of the word, it is more appropriate to design cases where the interaction is made with a moving object, for example, a ball.
Although it is extremely painful for the victim, it tends to be considerably funny to people who witness it.
Today in the baseball game the pitcher took a nutshot; the baseball hit him in the nuts.
Man, I just watched the funniest nutshot video ever.