How do I apply for a temporary taxation number as a US citizen who is an ordinary resident in Canada, living with my common-law partner?

If you are a non-resident of Canada and aren't eligible to obtain a Social Insurance Number, the following CRA link will take you to a T1261 form that is used to apply for an Individual Tax Number for non-residents.

T1261 Application for a Canada Revenue Agency Individual Tax Number (ITN) for Non-Residents