Gun Show Loophole Closing and Gun Law Enforcement Act of 2001 - Gun Show Loophole Closing Act of 2001 - Amends the Brady Handgun Violence Prevention Act to provide for regulation of firearms transfers at special firearms events (events at which 75 or more firearms are offered or exhibited for sale or exchange if one or more of the firearms has been shipped or transported in, or otherwise affects, interstate or foreign commerce, with an exception).

Prohibits a special firearms events frequent operator from organizing, planning, promoting, or operating a special firearms event without meeting specified requirements. Sets forth: (1) responsibilities of special firearms events infrequent operators, transferors other than licensees, and licensees; (2) special firearms event license application requirements; and (3) penalties for violation of this Act.

Provides an option for 24-hour background checks at special firearms events for States with computerized disqualifying records and programs to improve State databases.

Increases penalties for: (1) serious record keeping violations by licensees; and (2) violations of criminal background check requirements.

Gun Law Enforcement Act of 2001 - Authorizes appropriations for: (1) grants to States and local governments to support prosecutions in high gun crime areas; (2) establishment of up to 100 additional Project EXILE programs; (3) additional Bureau of Alcohol, Tobacco, and Firearms agents; (4) gun tracing and youth crime gun interdiction; and (5) grants to research entities developing technologies that limit the use of a gun to the owner.