The Body of Landru is the name inhabitants of Beta III used to describe all those under the control of Landru.
Six thousand years ago, the original Landru built and programmed a mechanical replacement to carry on his work. This machine managed the affairs of the entire planetary population; it may have qualified as a true artificial intelligence by some measures. At the time Kirk encountered it, it regarded the collective civilization of Beta III as a body. It is not known whether Landru programmed it this way, but considering his essentially benign goals, it seems unlikely. It seems more likely that the machine evolved this attitude over the centuries of its rule, as the most efficient way to implement the directives it had been given: protect the civilization and eliminate the evils of war, disease, and crime.
New individuals were added to the Body through a process called absorption. But not everyone on Beta III was part of the Body; some individuals existed as an underground, and opposed the will of Landru. Questioned about this, Reger explained that "some of us escaped the directives. Not many, but a few." Marplon later revealed that Reger was immune to absorption; it seems likely that this trait was shared by most of those in the underground. Another member of the underground, Tamar, was summarily executed by the lawgivers, after they were informed he had mocked them. Presumably, Landru took such mockery as evidence that Tamar was not a member of the Body. It remains unclear why it did not attempt to absorb him; the most likely answer is that it knew it had tried at some point in the past, and thus concluded that Tamar was immune to absorption. Such an individual would be a grave danger to it.
Landru used various other analogies to a body; it referred to those who created trouble as an "infection". It coldly described its plans to murder a large number of its citizens as "cleansing the memory of the Body". However, it never referred to the members of the Body as cells, instead using terms such as "members" or "parts".
Individuals within the Body existed in a kind of blissful trance, obedient to the directives of Landru, and subject to its direct control when contacted telepathically, a process refereed to as "summoning the Body". Landru could also focus its attention through specific members of the Body at need, but it clearly did not monitor all members at all times.
The lawgivers, Landru's footsoldiers, were an exception to this rule; they were under extremely deep control at all times, and they were able to initiate conversations with Landru when the need arose. (TOS: "The Return of the Archons")