Permanent resident of US or Canada, your question is not clear. What about citizenship?
Generally, as a US citizen or green card holder (permanent US resident), you are required to file a US tax return and pay US tax on all your world-wide income, no matter where you are living. You may also be required to pay tax in Canada. However, they are most likely not allowed to tax your US social security due to tax treaty provisions. And if they tax other income (like a pension or IRA withdrawals), you can usually claim an offsetting credit or deduction on your US return.
There are some variations, please let us know if you hold US citizenship or a green card.
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