The term "local government" is used collectively to describe many levels of overlapping government jurisdictions, such as
Different terminology is used in some states (for example, Lousiana has parishes rather than counties), but the general rule is that any government entity below the level of the state or territory is considered a "local" government.
Los Angeles is a unique metropolitan area, with both the city and county intertwined together. It can often be difficult to determine whether you need to work with the city government documents or county government documents to find the information you want.