A custom and a duty are essentially the same exact thing; they are both charges that are placed on goods that have been imported into a nation or items that have been brought into a nation by an individual. Often these two words are combined to make what is referred to as a custom duty.
A custom duty is a fee that is implemented under customs regulation and is collected before goods or items can be allowed legal clearance into the United States or various other nations. On goods, the proper paperwork has to be filled out regarding the importation of goods into the United States.
Once the paperwork is filled out, there is typically a duty or charge that is put on the bulk of the items. The charge on the item is dependent upon the type of item and whether or not there is an issue of domesticity of the item in the nation.
When the goods are shipped to the nation, a custom official goes over the shipment and collects the duty that has been assigned to it. From there, the goods have clearance to move into the United States and be distributed to the locations they are destined for.
In regards to custom duty regarding items being brought back by American citizens or visitors, these are items that require clearance as well. Certain food items, various plants and many other items that are acquired abroad incur fees when they are brought to the United States. However, there are certain items that are considered prohibited and these are immediately confiscated by customs.