Ex parte means on behalf of only one party (or one side) in court, without notice to any other party. For example, a request for a search warrant is an ex parte proceeding, since the person subject to the search is not notified of the proceeding and is not present at the hearing.
"Ex parte" is taken from Medieval Latin, this term is most often used in connection with legal procedure: an "ex parte proceeding" is one where only one side is present. These are dangerous proceedings because the other side is not there to present its side, and very one-sided rulings can result.
Latin terminology exists throughout the law, but legal procedure has fewer Latin terms than substantive areas of the law. "Ex parte," a procedural term, is a rare exception.