https://en.wikipedia.org/wiki/Form_4473

A Firearms Transaction Record, or ATF Form 4473, is a seven-page form prescribed by the Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF) required to be completed when a person proposes to purchase a firearm from a Federal Firearms License (FFL) holder, such as a gun dealer.[1]

Form 4473 contains the purchaser’s name, address, date of birth, government-issued photo ID, National Instant Criminal Background Check System (NICS) background check transaction number, and a short affidavit stating that the purchaser is eligible to purchase firearms under federal law.

The firearm dealer is required to record some information from the Form 4473 into a "bound-book", called an "Acquisition and Disposition Log”.[6] The dealer must keep the Form 4473 on file for the lifetime of the FFL, and is required to surrender the log book to the ATF upon retirement from the firearms business.

The ATF is allowed to inspect, as well as request a copy of, the Form 4473 from the dealer during the course of a criminal investigation. In addition, the sale of two or more handguns to a person in a five-day period must be reported to the ATF on Form 3310.4.

ATF form 4473 is de-facto registration of every legally owned gun in the United States of America