A counter hit is not a move or really anything you activate.
All it is, is when you hit someone during the start up of one of their moves.
Like say, in MK9, if you D1 someone just as they are starting the animation of their D1, your D1 would count as a counter hit. (though obviously it doesn't since MK doesn't have counter hits, this is just an example)
So if you D1 did like 8% damage normally, if it hits as a counter hit it would do 10% damage.
In some games, mostly 3D ones, sometimes moves have extra properties if they hit as a counter. Like say a normal that would normally just knock them away, if it hits as a counter would cause a launch state.
That's it, it's not a very complicated mechanic, or anything fancy. Just you do more damage if you hit someone in their start up frames.