The Brady Rule (or Rules) is the result of a U.S. Supreme Court ruling in the case of Brady v. Maryland (373 U.S. 83) in 1963. It held that the prosecution must disclose all exculpatory evidence to the defense counsel in criminal proceedings. The minimal requirements for such disclosures vary from state to state and at the federal level, but the common thread among them is that they relate to evidence that is material to either the defendant's guilt or the severity of the sentence (if found guilty). If a piece of evidence might sway the jury to find the defendant "not guilty" or might affect sentencing if found "guilty," then it must be handed over to the defense. 
Any time Jack McCoy or Casey Novak, for example, says that the defense gets to see their hand, as it were, but the prosecution doesn't get to see the defense's, that's a reference to the Brady Rule.
The state of Maryland prosecuted Mr. Brady and Mr. Boblit for first degree murder and the jury convicted both of them for the crime. Brady had insisted that he was involved but Boblit had done the actual killing, and while the prosecution had a statement from Boblit declaring just that, the prosecution hid that fact from the defense attorney when he requested Boblit's extrajudicial statements and got them both convicted for the murder with the death penalty attached.
The case was appealed to the Maryland Supreme Court, which ruled that the prosecution had violated Brady's right to due process under the 14th amendment to the U.S. Constitution, but that the effect of the statement would have been, at best, to reduce his sentence, not change his conviction; thus, he was entitled to a new sentencing hearing but not a new trial. The defense then appealed the case to the U.S. Supreme Court. In May of 1963, after hearing arguments over a couple of days two months earlier, they voted 7-2 in concurrence with the Maryland Supreme Court. 
The minimal requisite for what is considered "exculpatory" varies from state to state, but many share similar standards, including that:
- The prosecutor must disclose an agreement not to prosecute a witness in exchange for the witness's testimony.
- The prosecutor must disclose leniency (or preferential treatment) agreements made with witnesses in exchange for testimony.
- The prosecutor must disclose exculpatory evidence known only to the police. That is, the prosecutor has a duty to reach out to the police and establish regular procedures by which the police must inform him of anything that tends to prove the innocence of the defendant. However, the prosecutor is not obligated to personally review police files in search of exculpatory information when the defendant asks for it.
- The prosecutor must disclose arrest photographs of the defendant when those photos do not match the victim's description.
- Some state systems have expansively defined Brady material to include many other items, including for example any documents which might reflect negatively on a witness's credibility.
- Police officers who have been dishonest are sometimes referred to as "Brady cops." Because of the Brady ruling, prosecutors are required to notify defendants and their attorneys whenever a law enforcement official involved in their case has a sustained record for knowingly lying in an official capacity. 
Mentions by Name in Law & Order Episodes
- Season 5: "Competence"
- Season 7: "Good Girl" • "Showtime"
- Season 8: "Under the Influence"
- Season 16: "Kingmaker"
- Season 19: "Crimebusters"
- Season 20: "Dignity"
- Season 9: "Cold"
- Season 11: "Turmoil"
- Season 12: "Gray"
- Season 16: "American Disgrace"
- Season 17: "Criminal Pathology"
- The word "exculpatory" is a combination of the morphemes "ex-" ('out, from'), "culpa" ('fault', 'blame'), "-ate" (verb suffix), and "-ory" (adjective suffix); essentially, it means "having the quality of removing fault (or blame)" from someone.  In a legal context, it refers to criminal liability being shifted away from the defendant or defendants.
- While the U.S. Supreme Court ruling in Brady v. Maryland sets a standard for transparency in government, other states have had other cases that have set different, sometimes broader standards or affirmed the landmark ruling.