With a gift, the giver (not the receiver) is responsible for reporting it on their taxes, but only if the gift's value exceeds $15,000.
With an inheritance, the recipient may be responsible for reporting a portion of the inheritance. Usually, the inheritance itself isn’t taxable but any income produced as a result of the inheritance is for federal tax purposes. Some states have an inheritance tax. Refer to your state’s department of revenue to see what the requirements are for filing inheritance tax.