The buyer fills out the Form 4473 - this form has changed over the years, most recent now asks if buyer is hispanic.
http://www.atf.gov/files/forms/download/atf-f-4473-1.pdf
Then the dealer eyeballs it and makes sure the questions are all answered correctly, there are no abbreviations for dates, etc.
Then the dealer calls either a state department or the Feds for NCIC clearance. Here in Florida, they ask name, date of birth, social (if provided, it is still optional), and if it is a long gun, hand gun, or "other" (bare receiver). Then they put you on hold for a few minutes, polish their nails, talk to their coworkers, and eventually come back and giggle "whoops, forgot about you on hold". Then they use the magic words - "approved", "delayed" or "denied", followed by a long reference number. The dealer logs this response on the 4473, and then the customer leaves with the gun or comes back in a few days for it.
Approved - duh. Give the guy his gun and let him go on with his day.
Delayed - law says that if a sale is delayed, the feds have 3 (or 5?) days to get back to the shop to stop the transaction - if no word from them, the shop can transfer the gun or not, depending on shop owner policies. Many/most local gun shops will transfer at that point, most big box stores (wally world, cabelas, etc) won't.
Denied - someone has been a bad boy/girl. The customer can get a reference number, and with that contact the ATF/FBI to find out *why* they were denied - common names, close birth dates, etc. are causes for a lot of initial denials. If it becomes problematic, the buyer can apply for a UPIN, which involves finger prints and a longer FBI background check like getting a concealed carry permit. Once they have the UPIN, they can reference it on the 4473 and approvals happen.
On a fed level, *any* interstate transfer requires a FFL at the receiving end, and the 4473 and phone call. Some states require it as well for just pistols, and a few for all firearms.
Face to face sales are legal and unencumbered on a fed level, as long as both parties are legal to own and are residents of the state the transfer is taking place in. This is the "gunshow loop hole". All a gunshow does is get a bunch of like minded folks, some with cash, some with guns, all together in the same place. No different than putting an ad in your local classifieds, etc.
I'm not a FFL holder, but a buddy of mine is and he lets me call in the bg check for my own purchases.