This policy is often tied to what's called the "gun show loophole." Because the federal government doesn't require background checks for private sales, transactions at gun shows don't need to be regulated. A few states expanded the checks to all private gun sales, and a few added them for all sales of handguns.
Permit requirements vary by state. In California, for example, purchasers must get a Firearm Safety Certificate and show proof of residency -- unless the purchaser is active duty military -- to buy any kind of gun. In Maryland, purchasers must go through training and submit fingerprints, but only for handguns.
Some governments, like the District of Columbia, mandates that all newly purchased guns to be registered with a police agency. Others, like New Jersey and New York, only requires the registration of handguns. This generally means that the sale and serial number of a weapon is cataloged and that sometimes comes with a fee. Please note that there are a few states, like Virginia, that require registration only for "assault weapons," but that doesn't apply to most gun transactions.
Waiting periods are often implemented to give purchasers a "cool off" period, in case the gun is meant to commit a rash crime. The time varies considerably per state. Illinois mandates a 24-hour waiting period for long guns and a 72-hour waiting period for handguns. Hawaii mandates a 14-day waiting period.