Please, who is in US "co-op board" ???

JC, I actually think this article does a very good job of explaining what a co-op is. You need to read the entire article.

A co-op is a special type of real estate situation - the other owners in the building elect people from among themselves to serve on the co-op board. It is not ONE person; it is a group of people. They will determine whether someone is allowed to purchase a condo that is for sale.

You don't RENT when you're in a co-op. You buy. The co-op board will not collect your payments - that goes to your bank.

Please read the entire article and ask about things that confuse you.

EDIT: You will not be able to find a synonym. It's a very specific situation and there are not other words that mean the same thing.

Yes, it's owned by the people who own the apartments/condos in the building in the first place.

Let's say that there are 12 apartments, and six of them cost $2 million and $1 of them cost $1 million, so that's $18 million of value. Depending on the rules of incorporation, it could be that the people who own the $1 million apartments have a 1/18 share in the entire complex. Let's say that of the 12 apartments, they have 5 people on the co-op board. They will elect those people from among themselves. It will depend on what rules they set up when they created the co-op.