The Coalition for Human Longevity has existed in one form or another for nearly as long as humans have been in space. It exists to ensure the continuation of human existence and preservation of human history, science, art, and advancement. In order to achieve its mission, it has three branches:
The Coalition Congress: The congress is the primary law-making body of the coalition. They also oversee the allocation of funds and resources and approve new terraforming or space station projects. It is divided into two bodies, somewhat similar to the American Congress of ancient times: The Senate includes two representatives from each planet in the Coalition. The House includes proportional representation to the populations in each planet, as well as representatives from space stations and corporations relative to their population size. These representatives are voted into place by popular vote of their respective citizens.
The Coalition Judicial Branch: The Judiciary Branch of the Coalition is predominantly under the control of a complex AI system designed to evaluate Coalition laws, process evidence of wrongdoing, and issue punishment proportionate to the crime committed under the guidelines and parameters set by the Coalition Congress. It has also been programmed with a central code of human rights and will sometimes deem a law to be in violation of rights protections. Local planetary governments have their own judiciary policies and often have humans in charge of these systems. Generally speaking, law and the dispensation of justice is left to locals. The Judiciary Branch only gets involved where human disputes are appealed and require analysis without the interference of human emotional bias.
The Coalition Administrative Branch: The President and Vice President of the Coalition are voted in by the Coalition Congress--not by galactic popular vote. (This can sometimes be a point of contention.) The President and Vice President are primarily responsible for appointing leaders to the Departments of the Coalition and seeing to it that the departments are properly staffed and well-managed.
The Departments that fall under the purview of the Administrative Branch are more or less as follows (and this is not an exhaustive list because we'll add stuff if we determine it matters.
Department of Defense: This department oversees the Coalition Unified Defense Force. The Defense Force recruits from all planets in the Coalition (and any Coalition-affiliated space stations and corporation settlements). It's further divided into:
The Defense Security Agency (basically the spies and intelligence-gatherers of the Coalition)
The Planetary Defense Guard (basically security that stays stationed on planetary bases or space stations and defends them or provides aid in natural disasters)
The Coalition Defense Force (the main military force, based out of ships and space stations and roving to protect against pirates and other space threats--kinda the space navy. They are also involved in scientific studies and terraforming projects, especially in areas of space that could be threatened by raiders or independent military forces. If Picard's Enterprise were part of the CHL, it'd fly as part of the Coalition Defense Force.)
The Coalition Elite Corps (the highly mobile marine-esque strike teams).
Department of Education: This department provides accreditation for schools that operate throughout the Coalition. If you want to be sure your degree "counts" no matter what planet you go to, you want a DOE-approved school.
Department of Health: They do things like track diseases and help spread most current, most accurate scientifically-based medical knowledge.
Department of Justice: They oversee prisons and also investigations into crime and stuff.
Department of Planetary Development: They propose and monitor the development of new terraformed planets.
Department of Scientific Advancement: This department oversees branches of scientific work to benefit the larger coalition. These are things like agricultural development, conservation, weather-monitoring, exobiology/keeping an eye out for aliens, analyzing and predicting space phenomena, monitoring economics, conducting periodic census data collection, and so forth.
Department of the Treasury: They collect taxes and regulate banks and stuff.
Members of the coalition benefit from shared knowledge and mutual strength of the Coalition as a whole, but at the price of taxes that some people can sometimes feel are not fair. (Who ever thinks that taxes are fair, though?)