The Oligarchy is the collective term for the Oligarchs, a race of humans that have no corporeal bodies but are able to access almost any part of the Matrix. They have the ability to overwrite code, and can therefore defy laws of the Matrix which are otherwise rather concrete, even for advanced operatives such as Neo.
Oligarchs appear as glowing figures in the Matrix, and can appear with wireframe bodies, in the case of Halborn and Carlyne, or with more solid, uniformly colored appearances ala Helian and Tesarova. Due to their interface with the Matrix, the Oligarchs were impossible to attack using standard combat means such as martial arts or firearms, and were vulnerable only to hacking attacks.
All Oligarchs appeared to be human beings who had made a treaty with the Machines prior to the first Machine War in which they were able to program the machines not to attack them in exchange for Machine reign over the majority of Earth. The Oligarchy remained in an area commonly known as the No-Fly Zone on the Earth's surface, into which no Machines were permitted to enter. In addition, they had access to heavily fortified, EMP-proof hovercraft of unibody construction that could traverse the tunnels below the surface at rapid speeds.
Over time, the Oligarchs' physical human bodies began to deteriorate, and those who wished to live on passed their consciousness into robotic vessels. The Oligarchs, as competing individuals, rather than as a collective, sought out the Biological Interface Program in order to transplant their consciousness into sleeping bluepills in a process similar to that of an overwrite so that they could regain a human form.