The Core was a droid belief that all droids were connected, even though they came from different manufacturers and were of different degrees. Many thought this was a droid attempt to build a connection to the world around them, much like an organic being might feel connected to the ecosystem on their home planet. It was considered to be a hive mind, linking droids throughout the Galaxy.
Some droids thought it might be possible to manipulate their connection to the Core, much like Jedi and Sith use the Force.