Off-Balance sheet exposure refers to activities that are assets or liabilities of an entity which doesn't appear on the balance sheet. Example: Letter of the undertaking, Letter of credit etc.
A real-life example of off-balance sheet exposure is the Nirav Modi case. Letters of undertaking issued by Punjab National Bank were not mentioned as liabilities in the balance sheet of PNB. When Nirav Modi didn't pay his debts, creditors exercised the LOUs and Punjab National Bank had to pay for the debts taken by Nirav Modi. Although this liability wasn't mentioned in the balance sheet of PNB, they had to pay for it.