A Notice of Hearing is a prepared legal document that invokes all parties to hear a motion and may be emitted by any party. Most notably the notice contains a time and date for the court clerk to amend schedule for and what motion will be attempted. Motions require both sides to be present.
A petition is made to the court by a petitioner against a respondent, versus a complaint, which is filed by a plaintiff against a defendant. A petition asks the court to provide a court order, while a complaint is filed to seek damages or to get the defendant to start or stop doing something.