Countering is when a unit automatically attacks back with their weapon if attacked by a weapon. Counter-attacks can trigger enemy counters. The game will call it a "counter-counter" then "counter-counter-counter" and so forth.
Units have a "counter" stat which is the maximum they can perform per turn. Some units have zero by default, others higher. This stat can be increased in a number of ways depending on the game - passing Dark Assembly bills, Chara World , Innocents, etc. In every game barring the first release, Fist weapons raise the stat by two.
A unit's amount of counters for that particular turn is reduced by one by executing any action (attacking/skill use) with the exception of moving and lifting/throwing. Defending reduces your Counter amount by two, meaning that only units with Counter stat three or higher get at least one counter while defending.
In Disgaea 5, counter chance is 50% + remaining counter attacks x 10%.