Common to all is that the party making the threat will take some form of action of a legal nature.
Most common is the threatened initiation of a lawsuit against the second party.
Other threats might include an administrative law action or complaint, referring the other party to a regulatory body, turning the party into the legal authorities over a crime or civil infraction, or the like.
A cease and desist (C&D) letter is a formalized legal demand that a party stop ("cease") and refrain ("desist") from an activity that the demanding party finds objectionable, generally couched in formal language accusing the activity of violating the law.
[1] The objected-to activity may be most anything, although cease-and-desist letters are particularly common among certain areas of the law: A demand letter is a formalized demand by a party that another party pay money or take certain acts, often accompanied by a claim that the second party has engaged in illegal conduct, with an implicit or explicit threat that the demanding party will take some form of legal action.