"Petitioner" and "respondent" are legal terms used to describe the parties involved in a legal action. In civil cases, the petitioner is the party that instigates the legal action and the respondent is the person being sued. For example, in a divorce case, the petitioner is the spouse filing for divorce and the respondent is the other spouse. A defendant in a criminal case becomes a petitioner in some circumstances, such as if he seeks to have his conviction expunged.
In appellate cases, the party names work a little bit differently. The party appealing the judgment of the lower court is the petitioner and the party that prevailed at the lower court is the respondent, regardless of who filed the initial case. For example, if a defendant is convicted of a crime at trial and appeals, the defendant will be the petitioner and the government will be the respondent.