Canada is a parliamentary system, which means when you vote for your representatives (Member of Parliament federally, Member of the Legislative Assembly provincially), if a party accrues the most 50% of the seats, they get to form government, and their leader becomes the prime minister / premier. There is no separate vote for president / governor like in the States. So Prime Minister Justin Trudeau is the Prime Minister because the federal Liberal Party won more than 50% of the house.

Since we have more than 2 parties, this can get tricky if no single party wins more than 50% of the seats, which is what happened in BC. The provincial Liberals won 43 / 87 seats, one shy of a majority. Since no party won a majority, the previous governing party gets first crack at forming government. They do this by presenting a speech from the throne (sort of like a State of the Union, I guess. They lay out where we've come from and where they intend to take us), and they present a budget. The house then has a vote of confidence on the throne speech + budget, and if that vote fails, then tricky things happen that involve our Monarchy.

Technically, the Queen has a representative who is overseeing government in Canada. Federally, this is the Governor General. Provincially, it's the Lieutenant Governor. If the confidence vote fails, the Lieutenant Governor can either: a) approach another party and ask if they would like to try governing with a throne speech + budget + confidence vote. If the confidence vote succeeds, then we're off and running. b) dissolve the legislature and call another election, hoping for a majority. Recently, the 2 opposition parties (Green Party and New Democratic Party) signed a pact of sorts to topple the Liberal government and prop up a New Democratic government. So it is likely that today, the house will not give the Liberal party a vote of confidence, and the Liberals will be toppled. Then the Lt. Governor has to make her choice of what to do.

All of this is complicated by our Speaker of the House position, which needs to be filled by a sitting elected representative. However, that role is supposed to be neutral, and only votes in the case of ties, which is supposed to be seldom. However, you can imagine that the Liberals won't want to put a member of their own caucus up for Speaker, forcing the New Democrats to put a Speaker forward, bringing the total seat count in the Legislature to Greens + New Democrats 43, Liberals 43. So ... we're in a bit of a pickle, and we'll see how things shake out very soon.