The Core was the living heart of Ultraworld, a vast brain-like organism that, in the words of menial Relf, "lives to expand and must expand to live". Its custodians were the Ultra and used mindless menials to carry out physical tasks.
It expanded through the process of nucleoplasmic absorption, described by the Ultra as a very complicated process but essentially boiling down to consuming anyone unfortunate enough to be fed to the Core. Cally and Avon narrowly escaped this fate.
The power of the Core was considerable: Cally was transfixed by Ultraworld whilst still on the Liberator, it twice affected Tarrant, and was used to bring the Liberator into dock. The Core, however, could only comprehend logical thought, and it was distracted by Vila's illogical riddles and word puzzles while Orac reflected the Core's own wave emissions back at it, destabilising the Core and eventually destroying Ultraworld. (TV: Ultraworld)