A motion is a request to the court to enter a particular order. A motion for discovery is a request to the court to order the opposing party to produce discovery materials. Discovery materials differ depending on whether the pending matter is a civil or criminal case. Discovery is generally intended to allow parties to know what to expect at trial. Knowing the other side's case encourages pretrial resolution and facilitates the presentation of evidence at trial.
The local state or the federal Rules of Civil Procedure govern what material is discoverable in a civil case and how it can be discovered. In general, discoverable material in civil cases is any material that might lead to evidence that would be admissible in court.
In criminal cases, discovery is governed by the Rules as well as by caselaw. A criminal defendant is entitled as a constitutional right to discover anything the prosecution has that tends to show the defendant did not commit the offense charged, even if the prosecution does not believe the evidence. Many state and federal courts have open file discovery, which means that defense counsel can see the prosecution's file. Giving copies to the client-defendant, however, can be limited or even forbidden. (See e.g., local rules of the Federal District Courts in the Western District of North Carolina.)
Contrary to the original posting here, Grand Jury material is generally not discoverable. Grand Jury proceedings are sealed and confidential except in very rare circumstances.