A retaliatory act is one that attempts to get even with someone or to punish them for some wrongdoing. When you shove your brother because he shoved you, that's retaliatory.
Sometimes, wars are avoided because one country or group hesitates to attack another out of fear of a retaliatory strike. In fact, most wars consist of a long series of retaliatory actions — a bomb dropped to pay the enemy back for the bomb they dropped, and so on. Retaliatory comes from the verb retaliate, and the Latin root retaliare, "pay back in kind."