The Consensus makes all of the decisions regarding the security of The City, and public order. The head of the Consensus is The Speaker, who calls meeting in to order, and delivers the final say.
The Consensus was founded at the conclusion of the Faction Wars and was made to keep peace between the factions and the Vanguard.  They authorized the attempt to retake the Moon from the Hive, against Lord Shaxx's protests. After the invasion met with disaster, they also gave the order to retreat and collected intel and casualty reports.
At one point, Lysander and a group known as the Concordat attempted to rise up against the Consensus and the Vanguard, culminating in a pitched battle in a Tower that was later converted into a Crucible arena known as Bannerfall.