A core charge is an extra fee that you might have to pay when you buy certain parts for your car. It's like an extra deposit that you have to pay when you buy such things. The reason for this is because some parts (like batteries and starters) have a reusable part inside them that the manufacturer wants back.
So when you buy these parts, you have to pay the core charge fee. After you install the part and don't need the old one any more, you can bring it back to get the core charge refunded.