When a gift's value exceeds $15,000, the giver – not the recipient – is responsible for reporting it on their taxes. This is done on Form 709, United States Gift (and Generation-Skipping Transfer) Tax Return. Read more here.
For gifts valued at $15,000 or less, neither giver nor receiver need to report it.
Inheritances are usually not taxed on your federal return, but any income generated from the inheritance is (an example would be dividend payouts from stock you inherited). However, 6 states – Iowa, Kentucky, Maryland, Nebraska, New Jersey, and Pennsylvania – do collect some sort of inheritance tax.