An American who marries a Canadian citizen or permanent resident is eligible to become a permanent resident of Canada. Obtaining permanent residency in Canada via marriage is accomplished via the family sponsorship immigration route.
Spousal sponsorship is one of the available family sponsorship pathways. Spousal sponsorship is available for both legal marriages and common-law partnerships. The spousal sponsorship process can be started while the American spouse is already living in Canada or while they are still living in the US. Note that while the sponsorship application is being processed, the American spouse cannot work in Canada without a valid work permit.
The second part of the spousal sponsorship process is for the American spouse to apply for permanent residency in Canada. Becoming a Canadian permanent resident necessitates that the American spouse surrender their Green Card and permanent residency status in the US as it will be extremely difficult to comply with the residency requirements of each country’s permanent residency rules.
Once granted Canadian permanent residency, although you will be able to permanently live and work in Canada, you will not be able to vote in elections or run for political office. You may also lose your permanent residency status if you spend too much time outside Canada.
As a permanent resident of Canada, you eventually have the option to become a Canadian citizen. Becoming a Canadian citizen does not interfere with US citizenship as both Canada and the US accept dual citizenship.