The Core Worlds are the central planets within the Firefly universe.
As such, they are linked foremost by the cortex, an extremely advanced version of of the internet, so to speak. They also have the best technology, and are the oldest planets to have been terraformed within the universe, having fully established cities spread across their surfaces, along with their own sets of customs and rules (such as what constitues a reason for a life and death duel in Shindig) within those of their Alliance governmental parameters.
Being the most advanced, and wealthiest of the planets, the Core Worlds also have a high place in Alliance structure, being able to be resupplied of vital supplies in only hours (such as the hospital, St. Lucy's, in the episode Ariel), and being the seats of governing bodies, as we are told of things such as the Guild of Companions on Sihnon.
The two most prominent Core Worlds actually seen in the series are Ariel from Ariel, which, during exterior shots, looks highly similar to Coruscant from the Star Wars films. The second Coreworld shown in the show is Persephone from both Serenity and Shindig.