An adversary proceeding is a lawsuit arising in or related to a particular bankruptcy case. It is commenced by filing a Complaint with the bankruptcy court, and is given a separate case number. The Federal Rules of Bankruptcy Procedure and the Local Bankruptcy Rules state which actions must be brought as adversary proceedings, and the rules and procedures pertaining to adversary proceedings.
Both a creditor or a debtor may start an adversary proceeding, but they are most commonly started by creditors. A creditor might file an adversary proceeding to allow them to proceed with a foreclosure, or if it believes that a debtor has committed fraud.
Adversary proceedings can get complicated fast. Consider talking to a bankruptcy attorney the moment you get notice of one.