A company or government is deemed to be insolvent when it cannot pay its debts (i.e. is due to default). It is insolvency if it is due to not actually having enough money to pay off the debt, whereas simply not being able to access the money (which is a problem in itself) is a lack of liquidity.
When an entity is insolvent, some or all of its assets can be liquidated to pay off those with a claim on debt. Insolvency is usually undesirable for both the lender and the borrower as the assets are unlikely to be sold at face value, and therefore there is usually in the best interests of both parties to attempt to make alternative payment arrangements.