In common law legal systems, a motion to suppress is a formal, written request to a judge for an order that certain evidence be excluded from consideration by the judge or jury at trial. In the United States, the term "motion to suppress" typically encompasses motions in criminal cases where the proposed basis for exclusion arises from the United States Constitution, a state constitution, or a specific statute permitting the exclusion of certain types of evidence (for instance, a complaint that police procedures in a given case violated the defendant's Fourth Amendment right to be free from unreasonable searches and seizures).
A motion to exclude evidence where the proposed basis for exclusion arises from the rules of evidence is more commonly termed a motion in limine.
United States Law
In the United States, the motion to suppress stems from the exclusionary rule. As the court stated in Simmons v. United States: "In order to effectuate the Fourth Amendment's guarantee of freedom from unreasonable searches and seizures, this Court long ago conferred upon defendants in federal prosecutions the right, upon motion and proof, to have excluded from trial evidence which had been secured by means of an unlawful search and seizure."
Because it is grounded in the right to be secure from unreasonable searches and seizures, a person must have standing to move to suppress evidence. In other words, one cannot object to evidence obtained by an illegal search if it was someone else's privacy that was violated.
On a federal level, a motion to suppress is set down in Rule 41(h) of the Federal Rules of Criminal Procedure.