A formal, written document submitted to a court, and which asks for the court to redress what is described in the petition as being an injustice or an illegal act of some kind.
Petitions set out the facts, identifies the law under which the court is being asked to intervene, and ends with a suggested course of action for the court to consider (eg. payment of damages).
A person starting a claim by way of a petition usually takes the title of petitioner and the other side, a respondent.
Although procedures and terminology differ between jurisdictions, there are generally two ways to ask a Court to make a decision in a dispute.
One way is by petition which may restrict the evidence to affidavits only.
It's also a bit quicker and much less formal than the trial process.
Jurisdictions tend to limit petitions to situations which either rely heavily on documentary evidence, or where the fact pattern is usually straight-forward.
The other method is by "writ" and "statement of claim" which leads to a trial and which, with few exceptions, affidavit evidence is not allowed. This is usually the default way of engaging a Court to resolve a dispute.