The term "counter" is commonly used in 2 forms:
Counter Attacking between Mages: going to battle, again!
Counter is the act of performing a retaliatory attack against a mage that has hit you recently. The game allows to strike back a mage that attacked you for 24 hours time, regardless of common damage or pillaged Protection status (Exception: a mage in Council Protection cannot be countered).
Once a mage does more damage in taking his counters (combined army loss and land loss) than the enemy mage initially did to him, he no longer has the special countering privelages to pierce protection status. Also, the enemy mage gets counters on you for your counter-attacks, again lasting 24 hrs since your attack. Players decide to keep counter-counter-countering again and again, or follow some unwritten rule of one hit for one hit and no counter-countering... to not make too many enemies.
Example : John attacks Bob, doing 10% damage to his army, at 1200 hrs, January 1st. Bob now has a valid counter on John that lasts till 1200hrs, January 2nd. If Bob counters John at 2000 hrs, January 1st and does more than 10% damage to John, John gets a counter on Bob. However if Bob did not do more than 10% damage, John still gets a counter, but only AFTER 1200hrs January 2nd, and that counter only lasts till 2000hrs January 2nd(this counter is for the damage done by Bob's counter).
People use a wide variety of terms for the different types of counters: "Machine Counter" - the countering within 24 hrs allowed in the game, "Proxy Counter" - a counter taken by a guildmate or an ally rather than the defending mage, "Online Counter" - countering while a player is actively playing his/her mage, "Multi-Counter" - taking more than one "counter" for an attack.
Counter Attacking between units sent into battle: striking back
A unit which has "Counter Attack" (see Encyclopedia, Book of Units) will attempt to 'counter attack' whenever they are struck by an enemy's Primary Attack. When a unit attempts to use a 'counter attack' the unit will suffer from Fatigue, it will suffer Fatigue regardless of if the counter attack failed due to any of a variety of reasons (Paralyzed, Too Far Away, 0 Counter Attack AP, etc.)