This is a complex topic and anybody moving to the United States would be advised to seek professional help.
However, simply stated, one's liability to US tax is determined by whether one is resident or non-resident.
If one is resident, one's entire world-wide income is subject to US taxes. One may have tax liabilities in ones home country. If so and there is an appropriate tax treaty between the US and one's country of origin, then one's liability to home country tax can be offset against one's liability to US tax.
If one is non-resident then only the income arising from a US source is taxable
Whether one is resident or non-resident is determined by either
The Substantial presence test
The substantial presence test is met if the foreign national is:
physically present in the US for 183 days or more in the current calendar year or
is physically present in the US for more than 30 days in the current calendar year and his/her presence in the US over three years equals or exceeds 183 days which is based on the following formula
Green Card test means that a lawful permanent resident is classed as resident
from the first day he/she enters the US.
Tax treaties were in force on Jan 1st 1996 with the following countries: