A classical government is a type of traditional government that is based on beliefs or a structure that was present on the Old Earth. A country, such as Russia, or the United States, have founded new colonies on certain planets to continue on the structure of government that has served their respective countries since their founding.
Essentially, a classical government is a "continuance" of a government that had relocated from Old Earth to a new planet or planets. Many classical governments have had power shifts throughout the years, and where they are located, and how much area they claim to be theirs is constantly in flux.
Classical governments are typically operated in such a fashion that whatever operation was being put into use on Old Earth is transplanted to the new colony. Classical governments are slowly becoming the smaller percentage of governments in existence, however, they still hold the most power in the universe, as far as traditional governments go. While some contemporary governments may one day be as powerful as classical governments are today, it will be a long time before it happens.
Governments that ceased to exist at one point that are renewed in their founding are also called classical governments. For example, Prussia could be re-founded as a new government, and would be classified as a historically-relevant classical government, but a classical government nonetheless.