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.
Gifts valued at $15,000 or less don't need to be reported.
Inheritances are usually not taxed on your federal return, but any income generated from them (like dividend payouts from stock you inherited) may be.
Iowa, Kentucky, Maryland, Nebraska, New Jersey, and Pennsylvania collect a state inheritance tax.