What is the main purpose a government is serving?

Answer

The main purpose of a government is to control the activities and resources of a given place. Depending on the specific government, that control can be to the benefit of its citizens or to the detriment of its citizens. Most governments fall somewhere in between. A government can be the leadership of a tribal community or the power structure of China.

The most common form of controlling the activities of a people and a place is to enact and administer laws and regulations. Other activities may include (but are not limited to) administering the money; controlling both natural and produced resources; providing an armed force(s); a responsibility for the infrastructure of that place (water, roads, public facilities, etc.); some form of educational system; and providing a system and manpower to enforce its laws (which in some cases can be their armed force).

Answer (restated)

Any government exists to maintain an orderly society for its citizens. It does this by creating rules, laws, and legal systems to enable and regulate their interactions, and how they make use of the resources of the society.