Complaint
A claim explains what went wrong and demands compensation from the offending party, whereas a complaint explains what went wrong and merely demands correction or apology.
A complaint is the Pleadings that starts a case. Essentially, a document that sets forth a jurisdictional basis for the court's power, the plaintiff's cause of action, and a demand for judicial relief.
A plaintiff starts a civil action by filing a pleading called a complaint. A complaint must state all of the plaintiff’s claims against the defendant, and must also specify what remedy the plaintiff is seeking. After receiving the complaint, the defendant must respond with an answer.
Traditionally, in accord with federal courts' system of notice pleading, complaints (like other pleadings) did not need to be very specific, and did not need to specify the facts the plaintiff intends to prove. The Supreme Court introduced a heightened standard for complaints in 2007 with the case Bell Atlantic v. Twombly. This case requires that a complaint must allege "enough facts to state a claim to relief that is plausible on its face."
Complaints must be served on defendants. This lets defendants know that they are being sued and why. See Federal Rules of Civil Procedure; particularly Rules 3, 7, 8, and 10.