Two primary types of corporate bankruptcy. Chapter 11 and Chapter 7. Chapter 7 is a liquidation, where you have to sell all your assets and use the proceeds to pay creditors. This is what is generally referred to as "belly up", as it requires in most cases shuttering the business.

Chapter 11 is a reorganization of debt. Under the direction of the bankruptcy court a company works with creditors to restructure debt in a way that would enable them to pay it. It may include reducing interest rates, reducing the principle value of the debt or otherwise restructuring payment terms. Equity holders are often pushed aside in these proceedings. But operations continue and the business remains a going concern. This is what Hertz did.