The Citadel Council are the governing body for all species in on the Citadel. They maintain law and order within Citadel Space and have the final say on any issues brought up by the Ambassadors representing each species on the Citadel. Their are 3 species that have a seat on the Citadel Council. The Asari who are known for their diplomatic and mediation skills. The Salarians who are known for their intelligence and ability to gather information. The Turians are known for their military forces, their soldiers make up the majority of the Citadel Fleet.
To join the Citadel Council, a species must be able to provide the Citadel with resources such as a Fleet and economic aid. Species like the Elcor, the Hanar and the Volus who are physical unable to provide these resources are given an embassy instead. Those who represent their species embassy can tell their suggestions or complaints in front of the Council, however the Council still has the final word on any changes.
The Humans also have an embassy on the Citadel. Ambassador Udina is constantly pushing for the humans to get a seat on the Council. They can provide the Military and Economy power that the Council demand, however they are refused a place because they're still considered newcomers to the Galaxy.
The Citadel Council was formed around 3000 years before the setting of Mass Effect. The Asari were the first species to discover the Citadel. The Salarians were the next species to discover the Citadel, just a few decades after the Asari. The two species worked together to bring law and order to the Galaxy and to help expand their territory. The Asari and the Salarians were the only 2 species on the Council for the next 1400 years until the Turians were granted a seat for their contribution during the Krogan Rebellions.