What is the term for a legal action in which a person or group sues to collect damages for some harm that is done?

Lawsuit.

Explanation:

The word lawsuit is used in reference to a proceeding brought before a court of law where a petitioner (a party claiming to have inflicted loss because of a result of the actions of a defendant) demands a legal remedy. In other words, it refers to a case in which a court is asked to decide whether there is a disagreement between two persons or organizations. A lawsuit is a civil action brought before the civil court by one party against another.