US gun laws in a nutshell
Disclaimer:Generally, firearms cannot be legally transferred across state lines without involving an FFL.
For the benefit of the OP, since he's from another country:
FFL = Federal Firearms License, a federally-required license allowing the holder to conduct business across state lines. The U.S. Constitution gives the federal government the power to regulate interstate commerce. Due to some (controversial) court decisions that are beyond the scope of this discussion, this power has been extended to allow regulation of many transactions that take place entirely within a single state, so a FFL is always required to conduct business selling firearms in the USA (see below for more detail). FFLs are overseen by the Bureau of Alcohol, Tobacco, Firearms, and Explosives, often abbreviated as "BATFE" or simply "ATF".
There are various types of FFLs granted overseen by the BATFE, but the most commonly discussed types are the 01 "Dealer" license and the 03 "Curio & Relic" or "Collector" license. In short, the Dealer license allows businesses to sell firearms commercially, while the C&R license is given to individuals to allow them to buy and sell old, historical, and/or rare firearms out of state, including by mail order, but it does not grant any special privileges to buy and sell most modern commercially-offered firearms.
Dealer FFLs are required to conduct background checks on all buyers, whether the transaction takes place at a gun show or elsewhere. This involves collecting a form from the buyer, known as a Form 4473, certifying that he/she is legally eligible to buy the gun.
Nonlicensees- those without an FFL- and C&R licensees are not generally required to conduct background checks or collect Form 4473s. However, nonlicensees and licensed collectors are also prohibited from conducting business selling firearms, i.e. doing it for profit rather than as a hobby. Conducting business without a license is a federal crime. The problem is figuring out where the line is drawn.
OTOH individual states are allowed regulate transactions within the state's borders and may have requirements in excess of federal regulations.
Some states take this pretty far, requiring licenses, gun registration, and/or background checks for
all sales. Other states are fairly easygoing; for instance, TX law bars sales to anyone who is visibly intoxicated, under the age of 18, or the subject of a protective order, but face-to-face sales between individuals are otherwise allowed under most circumstances.