Solvency risk arises when a counter party is declared as insolvent and failed to honor the commitments and repay the bank as per the contractual agreement. Solvent parties have enough liquidity to honor all the commitments and at the same time they carry risk of becoming insolvent unexpectedly.
Solvency risk is higher if the liquidity position of the entity is tight and financial leverage is very high which can lead to even bankruptcy if the entity fails to honor its minimum commitment.
Suppose a bank has provided significant amount of loan to an aviation company with the contract to receiving the same within next 10 years along with the interest. Now if the aviation company fails to run afloat for next 10 years and declare bankruptcy in between, then the bank losses all the money it has lent to the company. For bankruptcy, the liquidation is done in which all of its assets are sold to repay the lenders first followed by preference shareholders and common equity shareholders. But for most the cases, the total amount from the sell proceed of the assets is pretty less compared to the original loan amount and the bank losses most of its money.
Banks can also face solvency risk from themselves where the banks become insolvent due to lack of liquidity in the system. In this case, banks may be sold to other banks or get government help to keep afloat.