Overview
A petition is a request to do something, most commonly addressed to a government official or public entity. Petitions to a deity are a form of prayer called supplication.
In the colloquial sense, a petition is a document addressed to an official and signed by numerous individuals. A petition may be oral rather than written, or may be transmitted via the Internet.
Legal
Petition can also be the title of a legal pleading that initiates a legal case. The initial pleading in a civil lawsuit that seeks only money (damages) might be called (in most U.S. courts) a complaint. An initial pleading in a lawsuit that seeks non-monetary or "equitable" relief, such as a request for a writ of mandamus or habeas corpus, custody of a child, or probate of a will, is instead called a petition.
Act on petition is a "summary process" used in probate, ecclesiastical and divorce cases, designed to handle matters which are too complex for simple motion. The parties in a case exchange pleadings until a cause for a hearing is settled. Black's Law Dictionary specifies it as an obsolete method used in admiralty cases. In the United States, the "act on petition" has been used in maritime cases.
From Wikipedia (CC BY-SA 4.0).