Under the NAFTA treaty, Canadians (as well as Mexicans) who practice certain professional occupations and who have a job offer from a U.S. employer may apply for TN status. It lasts for up to three years, and can be renewed in up to three-year increments, with no limit on renewals. For a list of the qualifying occupations, see the NAFTA Web page.
While in the United States on a TN visa, it may not be acceptable to apply for a green card. Some visas, such as the H-1B allow “dual intent”—that is, simultaneously assuring the immigration authorities that you are planning a temporary stay while also pursuing the possibility of a green card—but the TN is not one of them. Your best bet is most likely asking the employer for whom you are already working to petition for you to receive an H-1B visa, and then to seek permanent residence.