What is a corporation by estoppel and when does it apply?

Hypo 1: BadCorp enters into an agreement to buy 1,000 widgets from WidgetCorp for $10 each. A week later, BadCorp discovers it could have purchased the same widgets from another supplier for $5 each. While trying to figure out what to do, BadCorp discovers that it was improperly incorporated and is not actually a legal entity. This means that BadCorp's shareholders can save thousands of dollars by notifying WidgetCorp that BadCorp doesn't actually exist, and then choosing to create a new corporation that purchases from the cheaper supplier. WidgetCorp sues. Result: BadCorp believed and acted as if it were a legitimate corporation, so the court will not allow it to escape that fiction in order to screw over WidgetCorp and back out of their deal. Thus, BadCorp will be estopped from claiming it is not bound to the contract with WidgetCorp because it doesn't exist.

Hypo 2: PettyCo has been working with NiceCorp for years. One day, an employee of PettyCo stumbles across the fact that NiceCorp was improperly incorporated and not actually a legal entity. Upon hearing this, PettyCo attempts to sue NiceCorp's shareholders and hold them personally liable for a past due bill. Result: Assuming the shareholders were unaware of the defect, PettyCo may be estopped from holding NiceCorp's shareholders liable since it had always previously acted as if it were properly incorporated.