It depends. All gun sales have to go through NICS, the National Instant Criminal Check System. The exception is concealed carry permit holders in states where the holders meet certain criteria. I believe that criteria is that all permit holders in the state are actually run through NICS at certain intervals, but I am not sure.
NICS checks other databases for things like arrests, convictions, involuntary mental commitment etc that would make you a prohibited person. It can come back with a proceed, a deny, or a delay (which may fall into several sub categories). Typically, if you get a delay, the store is allowed to transfer the firearm to you after 3 days, meaning they typically have that amount of time to investigate it further before the store can give it to you and they may have to take action to get it back if the check comes back denied. This is what happened in the Roof shooting in Charleston. I worked at a big chain store (Academy) for a while, and our policy was not to transfer any guns without a clear proceed, even if the law said we could transfer on a delay. Stores can sell to whomever they want based on their own conditions, and that was one of ours.
Guns are not registered. The information is collected for a check, but this information only includes your identifying information and the type of firearm(s) (handgun, long gun, other). The information, including manufacturer, model, caliber and serial number is recorded on the form 4473.
The form 4473 has your identifying information, physical attributes, legal questions (are you a felon, do you use illegal drugs, have you renounced your citizenship, do you have a protective order against you etc), attestation that you've provided correct information, seller's information etc. This is not sent to an agency or database, but has to be kept with the dealer for 20 years and may be viewed as part of a regular inspection or if needed in connection to a crime.
Here is an article I wrote explaining some about the process
http://www.goodgamehunting.com/where-does-your-information-go-when-you-buy-a-gun-in-the-us/
Indiviso states may have their own laws including background checks, registration, ownership licenses, licenses to buy ammo etc but they are in the minority. Most states are not significantly stricter than federal law