The Queen of Canada has very little true power. She can theoretically strike down bills passed by parliament but in reality grants all of her power to the Governor General of the country and the Lieutenant Governors of each of the provinces. It's not clear whether, if she chose to disagree with a provisional law, her opinion would really even be considered. In practice, Canada is a constitutional monarchy.
The Queen of Canada is a position different from the that of the Queen of Great Britain and Northern Ireland. Queen Elizabeth II holds both titles but she is the Queen of Canada as a result of Canadian law, not UK law. Canada and the UK are 100% independent of each other and both have equal standing in the Commonweath (of which she is also symbolic head).
To quote Wikipedia:
Elizabeth II (Elizabeth Alexandra Mary Windsor) (born 21 April 1926) is Queen of sixteen independent nations known as the Commonwealth Realms. These are the United Kingdom of Great Britain and Northern Ireland, Canada, Australia, New Zealand, Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, the Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Antigua and Barbuda, Belize, and Saint Kitts and Nevis. By the Statute of Westminster 1931 she holds these positions equally; no one nation takes precedence over any other.