Concealed carry permits let you carry one that isn't visible. And getting that permit is necessary in every state that allows concealed carry.
In several states, you theoretically don't need a permit to open carry.
In many states, you need a background check to buy a gun from a licensed dealer - but not necessarily one to buy from an individual in private sale.
The licensing rules vary from state to state, but the short answer is - yes, you need at least a background check to buy one and/or carry one in most states.