Some Commenwealth countries recognize the Queen as head of state — many don't.
Canada, lacking a revolution, has retained ties to the Queen. We went through a lengthy (and ongoing) evolution — for instance, until our Constitution was “patriated”, the final arbiter of some constitutional matters was the “Law Lords” in the British House of Lords.
The Queen is Canada's Head of State, while the Prime Minister is our Head of Government. Your President (who shall go unnamed) serves both roles. Our Governer General is the Queen's representative in Canada, and serves as Head of State in the absence of the Queen (which is most of the time).
The Queen (and GG) is a titular head; but with many reserve powers. (Even constitutional lawyers couldn't tell you exactly what those are.) She's also the symbol of the country — just as the flag is. We tend to place a lot less importance on the flag as a symbol, compared to you Americans — because the Queen, or GG serve that function too.
Of course, this is all changing — and the role of Royalty will probably take a big hit after our current (very popular) Queen passes.
Thanks for your explanation of the Queen and Canada. So she really is your Queen. Cool! (I know, I’m a Yank and tradition (here anyway) says we should not look toward the monarchy with favor, but I do anyway. Such cool houses! the jewels! the TRADITION! I like it!)