Piercing the corporate veil (PCV) is the technique a creditor will use to attempt to reach the personal assets of a business owner whose business interest is on the wrong side of a judgment. In the context of small business owners, its a single member business or partnership that can't satisfy a debt or judgment incurred by the business with business assets. In such an occasion, assuming the debt or judgment is valid, the creditor is stopped when all business assets are liquidated. That is the benefit of the corporate (of LLC) form. It is a completely separate entity from the individual owner. However, if the creditor can prove the owner(s) abused the corporate form, the creditor may succeed in piercing the corporate veil and reaching the individual's personal assets to satisfy the debt. It's a worst case scenario for a business owner, but it can be avoided if specific formalities are followed, the entity is adequately capitalized and the entity is formed, funded, and carried on for a legitimate purpose.
First, don't get lazy. If you don't respect the division between yourself and your business, you can't expect a creditor to either. If you run a corporation, maintain annual meeting minutes, keep an accurate stock ledger, file your annual reports with the Secretary of State and pay your fees. An LLC requires less, but it's recommended to maintain similar administrative habits. Even more critical is keeping business bank accounts and expenses separate from personal debts. If you treat the business account as your personal piggy bank, forget about a court respecting the corporate identity.
Next, insure the business. That may include insuring the property, product liability insurance, and professional and general liability policies. A commercial umbrella policy doesn't hurt either. In the same way you insure your own car and home from liabilities that may arise, you must think the similarly about your business. Undercapitalization invites a successful veil piercing action. Adequately capitalizing your business such that it can reasonably meet prospective liabilities is a must and insurance is a terrific way to accomplish this.
Finally, forget about a court of equity respecting the corporate form when it was opened or funded for unethical or dishonest purposes. This may be a no-brainer, but stuffing real property or cash in an LLC to hide it from known creditors is never going to work.
When successful veil piercing happens, frequently it's the result of a lackadaisical, ignorant or dishonest owner treating the business as an alter ego of himself. If you think of the business as a distinct and separate individual from yourself and deal with it as such (don't steal from its accounts, contract with it at arms length, etc.), a lot of the mistakes that lead to veil piercing can be avoided.