Insolvency is the inability to pay debts.
Bankruptcy is one way of dealing with insolvency.
Bankruptcy in the balance sheet sense exists when a debtor's debts exceed the debtor's assets. Bankruptcy in the equity sense exists when the debtor is not able to pay debts as they fall due. The concept of insolvency is important because several remedies inside and outside of bankruptcy hinge on the insolvency of a party. For example, outside bankruptcy a buyer of goods may be able to recover a down payment for those goods if the seller misrepresented its solvency while the seller was insolvent. Similarly a seller of goods can recover those goods if the buyer misrepresented its solvency while actually insolvent.