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You issue a debit note to someone who owes money to you i.e. debit note is the same as an invoice. Example: You supplied a product or service to a customer and issued him a debit note. A customer would settle the debit note by usually paying you the amount as per the debit note.
You issue a credit note to someone whom you need to pay. Example: A customer paid you for 100 units of a product but you supplied them with 50 units. You would then issue a credit note to the customer for an amount equivalent to the undelivered 50 units. The credit note can be settled by either paying back the amount to the customer or adjusting the amount against supply of other goods/services to the same customer.