The Brady Handgun Violence Protection Act, often called the Brady law, was part of President Bill Clinton's Violent Crime Control and Law Enforcement Act of 1994, according to Wikipedia. The law calls for a five-day waiting period after purchasing a firearm and a background check. The law was popularly called the Brady Bill, because James Brady campaigned hard for it. Brady, a White House secretary, was shot during the 1981 assassination attempt on Ronald Reagan.
According to Wikipedia, Clinton signed the Brady law on November 30, 1993. It came into effect on February 28, 1994. The law requires that before someone purchases a gun, he must get approval from the National Instant Criminal Background Check System (NICS), a database maintained by the FBI. So long as there are no further state-level gun requirements, if NICS grants approval, then the buyer can purchase a gun from federally approved dealers.
Certain gun purchases do not require NICS approval, namely those purchased under a Curios and Relics license. These guns must be at least 50 years old, certified by a curator, or possess some monetary value based on rarity, strangeness, or association with a person or time period.