What the heck is a ‘dominion’ anyway?

This weekend I had a terrifically long conversation with my mother about the British North America Act, the precise nature of the relationship between Canada and Great Britain (read: the United Kingdom) and the Commonwealth of Nations, and the proper name for the country. The last common one that I know of is ‘Dominion of Canada’, but I haven’t heard anybody ever refer to it that way—at least not ironically. Of course, the country is referred to as a ‘dominion’ in a number of different contexts even today, yet by and large these are archaic, such as, perhaps most famously, the third verse of the original text of ‘The Maple Leaf Forever’:

Our fair Dominion now extends from Cape Race to Nootka Sound,
May peace forever be our lot and plenteous store abound,
And may those ties of love be ours which discord cannot sever,
And flourish green o’er freedom’s home the Maple Leaf forever!

But as far as most people who live here are concerned, I believe, the name of the country is simply ‘Canada’. However, on landing at YVR today, I was officially welcomed by the (Los Angeles-based) flight crew to some place called the ‘Republic of Canada‘, much to the amusement of about 75% of the plane. I suspect that this is not the short-lived Republic of Canada from the 19th century, but rather some magical mystical country without a Queen and an absurdly long list of successors.