A bad debt is an account receivable that has been clearly identified as not being collectible. This means that you remove that specific account receivable from the accounts receivable account, usually by creating a credit memo in the billing software and then matching the credit memo against the original invoice, which removes both the credit memo and the invoice from the accounts receivable report.
When you create the credit memo, you credit the accounts receivable account and debit either the bad debt expense account (if there is no reserve set up for bad debts) or the allowance for doubtful accounts (which is a reserve account that is set up in anticipation of bad debts). The first alternative for creating a credit memo is called the direct write off method, while the second alternative is called the allowance method for doubtful accounts.
A doubtful debt is an account receivable that might become a bad debt at some point in the future. You may not even be able to specifically identify which open invoice to a customer might be so classified. In this case, you create a reserve account for accounts receivable that may eventually become bad debts, estimate the amount of accounts receivable that may become bad debts in any given period, and create a credit to enter the amount of your estimate in this reserve account, which is known as the allowance for doubtful accounts. The debit in the transaction is to the bad debt expense. When you eventually identify an actual bad debt, you write it off (as described above for a bad debt) by debiting the allowance for doubtful accounts and crediting the accounts receivable account.
For example, ABC International has $100,000 of accounts receivable, of which it estimates that $5,000 will eventually become bad debts. It therefore charges $5,000 to the bad debt expense (which appears in the income statement) and a credit to the allowance for doubtful accounts (which appears just below the accounts receivable line in the balance sheet). A month later, ABC knows that a $1,500 invoice is indeed a bad debt. It creates a credit memo for $1,500, which reduces the accounts receivable account by $1,500 and the allowance for doubtful accounts by $1,500. Thus, when ABC recognizes the actual bad debt, there is no impact on the income statement - only a reduction of the accounts receivable and allowance for doubtful accounts line items in the balance sheet (which offset each other).
Thus, a bad debt is a specifically-identified account receivable that will not be paid and so should be written off at once, while a doubtful debt is one that may become a bad debt in the future and for which it may be necessary to create an allowance for doubtful accounts.