Core size applies to labels that come in roll format. The core size refers to the diameter of the plastic or cardboard tube on which your labels are rolled.
The inside diameter (I.D.) will determine if the label roll will fit on the label spindle, and as such is compatible with your printer. The outside diameter (O.D.) is less important, as it plays a role in determining the length of the roll (i.e. number of labels the roll can hold).
Most desktop thermal printers are designed to work with 1” (25 mm) cores, with the bigger, more industrial models set-up to work with 3” (76 mm) cores. 2” (51 mm) cores are less frequently used, but typically roll printers for inkjet labels will use this size.