claim (prayer or demand) for relief – a complaint and any pleading which sets forth a claim, counterclaim, cross-claim, or third-party claim.  — aka demand for relief; relief; prayer for relief.

counterclaim – a cause of action within the defendant’s answer which may have been used to sue the plaintiff in a separate action; used in opposition to or as a setoff against the plaintiff ’s claim.

• compulsory counterclaim – if a defendant fails to assert a compulsory counterclaim in the original action, that claim may not be brought in a later, separate action.
• permissive counterclaim – does not arise out of the same subject matter as the opposing party’s claim or involves third parties over which the court does not have jurisdiction, and may be brought in a later, separate action.

cross-claim – a claim asserted against a codefendant(s) or coplaintiff(s) in a case and that relates to the subject of the original claim or counterclaim. See Fed. R. Civ. P. 13(g). — aka cross-action; cross-suit.

third-party claim – a claim to property held by the defendant, set forth in a third-party complaint.