What are the process for Canadians to marry a US Citizen?

The process for Canadians to marry US citizens is the same for natives of any other country marrying a US citizen. You can marry in either a civil or religious ceremony. If in a religious ceremony, you would show USCIS that the marriage is registered by the state and not just registered in your faith. If in the US, you could be married here if you desire and if you are in Canada, you could marry over there. Both parties must be present at the time of the marriage to make it legal. In the US, you could apply for adjustment of status to permanent residence without leaving the country assuming that you entered the US legally and have no other impediments to permanent residence. In Canada, your spouse could file an I-130 relative petition with the USCIS lockbox in Chicago, Illinois, and after approval, you would undergo consular processing through the National Visa Center and ultimately interview at the American consulate in Montréal for your immigrant visa. Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.

It easiest and fastest to bring your girlfriend to the US on a fiancé visa and get married in the US, after which she can apply for a green card from here in the US.