It's hard to find a complete equivalent in the US system. The PM is simply the head of our government. The head of state is the Queen, as represented by the Governor-General. All almost entirely ceremonial as the G-G has no real power. They just agree with the government. Our parliament is bicameral, like yours, but the Senate is also largely ceremonial, and except for their involvement in endless scandals over personal expenses, the senators do little but rubber stamp legislation passed by the lower house(the Commons).
The Prime Minister is essentially the majority leader, except that since there's only one House that has any power, there's only one majority leader. He or she is picked internally, generally ahead of an election, in a system that somewhat resembles closed primaries. If one has a party membership, one can be involved in the process. Unlike in the US, our federal and provincial political parties don't always match up. Some parties on the federal level have no branches in certain provinces(for example, our left wing NDP party does not have a local affiliate in the largely francophone province of Quebec), while some provincial parties have no federal equivalent(like Saskatchewan Party, currently ruling that province). Even more confusing is that two provinces have a Liberal Party that is not affiliated with the federal Liberal Party. In the case of British Columbia, the provincial Liberal Party is actually on the opposite end of the political spectrum from the federal Liberal Party.
Long story short, the PM is the leader of the governing party, and as such, can be replaced at any point by his or her party at any given point, with no need for an election. I don't recall a time when this happened in Canada, but it did happen recently in Australia, who have a very similar system to ours. The PM at the time, Tony Abbott, lost the support of his own party, who opted to replace him with a rival, Malcolm Turnbull. This was a vote inside the party, and the change happened without a new election.
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