What does it mean if a government is centrally controlled (APUSH)?

It means all the power rests in the federal government, as is the case in a traditional kingdom, like Saudi Arabia, or a dictatorship. The US is unusual in the sharing of power between the federal government and the states. For example, even the legal code varies from state to state, with some states having the death penalty for certain crimes while others don't. In many countries, the laws are made by the central government and are the same all over the country. In the US, there's a constant tension between what the states control and what the federal government controls. In fact, the states are called "states" (a word that in other contexts is synonymous with"country") because the founding fathers saw the US as a looser organization of independently controlled states than we currently have.