If you are a US citizen or permanent resident (green card holder) you must file a tax return to report all your world-wide taxable income. Gifts are not taxable income, but you may have other taxable income such as bank interest, investments, or money from side jobs. Even if you file and pay tax overseas, you must also file and pay tax to the US, although the US will generally give you a deduction or credit if you pay foreign tax on the same money.
In addition, if you are a US citizen or green card holder, you must report gifts from foreign persons of more than $100,000 and gifts from foreign corporations of more than $16,000 using form 3520. No tax is owed but there is a reporting requirement, and there can be a penalty if the form is not filed when required. Form 3520 is not part of your regular tax return but it is filed with the same deadline (April 15, 2024 for gifts received in 2023).
Lastly, if you are a US citizen or green card holder, and at any time during the year you own or control a foreign bank account with more than US$10,000 (or equivalent) funds, you must file a report called FBAR, this can be done electronically from the IRS web site. Again, no tax is owed, but there can be a penalty if you don't file when required. (You don't need to report owning a US bank account no matter what the balance is.)
And really last, if you are NOT a US citizen or green card holder, you are only required to report and pay tax on income that is "effectively connected" to the US. That could include selling property in the US, or working in the US for a few weeks while you are traveling around, but "effectively connected" does not include owning stocks in US companies or receiving a gift, even if it is deposited in a US bank account. (If the bank pays interest, the interest is effectively connected, of course, but not the principal.)
