What is a lieutenant government?

I would assume you mean "lieutenant governor". A lieutenant governor can be two things. A lieutenant governor in the US is like a vice president to a governor. They don't really do anything; but they take the governor's place if he can't complete his term. They are often elected together with the governor, like the president and vice president are, but sometimes are chosen separately. The latter can result in weird dynamics - republican governor, democratic lt. governor.

In Canada, a lieutenant governor is the queen's representative in each province. Each Canadian province has its own "prime minister", called the premier, and the dynamic is kind of like that of the queen to her PM in that the lieutenant governor has no power over the government but is kind of a ceremonial figure.