More properly named the The Galactic Federation of Free Alliances, and also known as the Galactic Federal Republic, The Galactic Alliance is the current incarnation of the New Republic, so-named when then-Chief-of-State Cal Omas determined that a new government was necessary to address the threat of the Yuuzhan Vong War after the battle of Ebaq 9, widely believed to be the turning point of the war. The new Galactic Alliance was composed of the systems governed under the New Republic, what remained of the old Galactic Empire, and the Hapes Consortium.
Initially, the Alliance was headed by the High Council, which advised and administered for the Chief of State. This body was composed equally of Jedi and Alliance officials. However, this arrangement did not last long, as only 8 years later, following the incident known as the Swarm War, Grand Master Skywalker dissolved the Jedi role in the Council, believing Jedi should not have such a direct role in politics. After this, the Council was replaced by the Advisory Council, a body originally composed of senior Senators tasked with non-military advice and administration, now given a larger role in galactic government.
More recently, the top executive body has undergone yet another transformation. In an effort to implement better checks and balances on the senior leadership of the Alliance, following several embarrassing overreaches by successive Chiefs of State, there is no longer a single leader of the government, but instead a Triumvirate. All executive decisions are made by a majority vote amongst these three figures.
Considering all these changes to the executive branch, the legislative body has remained relatively unchanged. Since the Republic’s reformation into the Galactic Alliance, the Senate has had little to no executive role in government, but remains the primary body for the passing of all legislation.
The judicial branch, too, has been fairly stable in the Galactic Alliance. Indeed, it saw the least substantive change from the New Republic. The Court of Justice is narrowly concerned with violations of the Galactic Alliance Common Charter, the constitutional document of the Alliance, by the other two branches of government. Unlike its analogue in the Old Republic, the Supreme Court, the Court of Justice is not the “court of last resort” for ordinary civil and criminal cases—it only concerns itself with Charter-based cases. Lesser matters are handled by regional courts instead.