United Earth

The Seal of United Earth

United Earth is the name of the planetary nation-state created through the unification of Earth in the 22nd Century following First Contact in 2063. United Earth later continued to exist as a political subdivision of the United Federation of Planets when its government helped found that nation in 2161. (Star Trek: First Contact, Star Trek: Enterprise.)

The last of Earth's early nations became a part of United Earth in 2150. (TNG: "Attached") A forerunner of the United Earth Government was the European Hegemony which existed in the 2120s. (TNG: "Up the Long Ladder")

Common misbelief is that Australia was the last nation state to join the United Earth Government. However under close observation the given line in "Attached" does not support that claim. It is, however, possible that the United Earth government was founded prior to 2150 and that this was the year that the last remaining independent nations, including Australia, chose to join. This is supported by various episodes of Star Trek: Enterprise, such as "First Flight," which established that the United Earth Starfleet existed in the 2130s and 2140s.
The Official Star Trek Website (The official Star Trek-website) claims that 2113 may be the year that United Earth came into being, this fact has never made it into the canon.

Nation-states and confederations, such as the United States, retained their individual identities after 2150, however. The European Alliance, African Confederation and others continued to exist into the 24th century and beyond. (ENT: "Affliction"; TNG: "Conundrum")

Presumably, the United Earth Government at the time of Star Trek: Enterprise (who is responsible for the Earth Starfleet) will remain in place following the founding of the United Federation of Planets. The 23rd century existence of a United Earth Space Probe Agency in TOS Season 1 indicates this name persists referring to the world government even after it joins the UFP.
The exact structure of the United Earth Government is unknown. Nathan Samuels, an Earth diplomat who appeared in several episodes of Star Trek: Enterprise, is referred to as a Minister, suggesting that Earth has a parliamentary system of government. Thus, the leader of Earth would be a prime minister, perhaps Samuels himself.