I am a graduate student at Cornell University in the US and I receive a scholarship (NSERC) from the government of Canada (~$20,000). Do I declare the scholarship to the US government, Canadian government, or both?
If you're a Canadian citizen studying on a student visa in the US, then your scholarship given to you in Canada is not taxed in the US (that's according to the US-Canada tax treaty, article 20). There are more detailed explanations and examples in the 2008 protocol (article 15 of the protocol).
In order to claim treaty benefits, you must attach form 8833 to your US non-resident tax return (I suggest doing it through a professional tax adviser, not on your own).
You should consult a licensed tax professional (EA/CPA licensed in the US), and a Canadian tax accountant to get a more qualified answer to this question, and to learn how scholarships are taxed by Canada.