De facto corporation and corporation by estoppel
Okay, so imagine you and your friends want to start your own club. You have rules you made up and you all agree to follow them. But you forgot to officially register your club with the government. Even though you aren't a "real" club yet, you act like one, you follow your own rules, and people think of you as a club. This is kind of like a de facto corporation - it's not officially registered, but it acts like it is.
Now imagine one day someone wants to sue your club for something. Even though you're not a registered club, the court might still treat you like one because you've been acting like one. This is a corporation by estoppel - it's not officially registered, but people are treating it like it is.
Basically, a de facto corporation is a group that has been acting like a corporation, even though it's not officially registered. And a corporation by estoppel is a group that people treat like a corporation, even though it's not officially registered.
Now imagine one day someone wants to sue your club for something. Even though you're not a registered club, the court might still treat you like one because you've been acting like one. This is a corporation by estoppel - it's not officially registered, but people are treating it like it is.
Basically, a de facto corporation is a group that has been acting like a corporation, even though it's not officially registered. And a corporation by estoppel is a group that people treat like a corporation, even though it's not officially registered.