Unified Earth Government

The UEG is the central civilian government for Earth and the innermost colonies. The UEG houses the Colonial Administration Authority, and therefore the UNSC.

Differences Between the UEG and UNSC
The UNSC is the exploratory, scientific, and military arm of the human government. The UEG is resonsibile only for Earth and the Inner Colonies, the main locations of human government. The UEG would house the executive, legislative, and judicial branches of government, leaving defense mostly the responsiblity of the UNSC. It may act as a sort of United Nations, bringing Earth nations together into a unified government.

UEG Military
In the Halo: Combat Evolved manual, it mentions UEG ground forces under the Warthog section. The UEG's military is probably a reserve defence/peacekeeping force. It may be that the UNSC Army is actually the UEG Army. In the opening cinematics of Halo 2, it is revealed that the UEG exploratory/ military organization is dubbed the UESC, or United Earth Space Corps.

Marathon
It should be noted that both the UEG and UESC are references to the human government and military in the early Bungie game, Marathon.