Canada is a Constituional Monarchy... The Queen is still, in essense our head of state. A figure head in the truest sense but still the head of state.
All laws enacted by the Parliament must be signed by the Govenor General (essentially a rubber stamp). The Govenor General is appointed by the Prime Minister and is the Queen's representative on our soil.
As for Socialism... Yes, as a nation we tend to lean left of the US on most social policies. We believe in looking after our citizens by way of a social saftey net. No, it isn't always effecient but we (as a nation) feel it is important so we continue to do it.
However, on the whole we are way more right leaning than say Cuba, Sweden or many communes in the lower 48...