It depends on whether the company is a limited liability entity (LLC or Corporation), and also depends on what kind of liability is at issue.
Both Limited Liability Companies ("LLCs") and Corporations (both "S-Corps." and "C-Corps.") enjoy "limited liability." This means that the owners of the entity ('members' in the LLC and 'shareholders' in the corporation) are not personally liable for the debts of the entity. So if the entity suffers a loss (defaults on a debt, or is found liable for a tort claim), the creditor is only entitled to recover against the assets of the entity (so any assets that belong to the entity, or any insurance policy held by the entity).
There are 3 exceptions: (1) if the owner has agreed to act as a "personal guarantor" (usually you will see this with a lease, where the landlord requires the owner to guarantee the lease terms in the event the entity defaults); (2) if the owner causes their own tort (we are always liable for our own torts, for example, if the owner is driving a company car and hits a pedestrian, both the owner and the entity are liable); and (3) "veil piercing" (this is not very common, but can happen if the entity is "undercapitalized" - discussed in more detail below).
Veil piercing is used to compensate an injured party that is harmed by a company (either LLC or Corp.) that is inadequately capitalized (or even less commonly, due to "irregularities" in corporate structure - but this is largely disregarded). What happens here is when the entity does not have enough money or assets to cover the type of risk created by the activity they are engaged in. A good example is a taxi company that doesn't have any assets (it leases a car, has no accounts receivable, no employees, and no insurance). It simply doesn't have sufficient assets to cover the type of risk it is placing on the general public (driving around mid-town with pedestrians and other vehicles everywhere). If another person is injured, they will be entitled to sue the entity's owner directly (via "veil piercing") in addition to the entity because there was inadequate capitalization. This problem can be easily fixed, if the taxi company in our example simply took out a reasonable insurance policy, they could cover foreseeable losses and the owner would not be subject to liability (they have "capitalized" their company).
(You also see veil piercing used to prevent owners from using their companies to perpetuate fraud on third parties - so you cannot create an LLC or Corp. to defraud people and then hide your own assets, think Bernie Madoff).
This limited liability only works "one way" - you can protect your personal assets from debts of the LLC, you do not protect your assets of the LLC from your personal debts. Many people get confused with limited liability, and believe that if they set up a separate entity, they can protect assets from their personal debt collectors (so they can shield their personal assets from their personal debts by placing their belongings into an LLC). It doesn't work this way. If a creditor has a judgment against an owner, they will simply take the owner's interest in the entity (so a member's interest in the LLC, or their shares of the corporation), and in this way take control of the individual assets.
If this is not a limited liability entity - so it is either a "sole proprietorship" or a "partnership" (or an "unincorporated entity"), the owner will be personally liable for all liabilities and debts of the company.