A coop is a not-for-profit corporation that is composed of shares, like shares in a commercial corporation. Lets say that the coop has divided itself up, what they call "offered", 64,000 shares. The complete value of the coop is represented by those 64,000 shares. Lets say that the only thing the coop owned was a building, worth $64,000. So, each share would be worth a dollar.
So, the coop divides up its apartment building by assigning numbers of shares to each apartment. Since they've issued 64,000, that's what the total has got to be. Each apartment then is related to a certain number of shares. Obviously, bigger apartment, more shares. Better apartment (like, higher floor, or, with balcony), more shares.
So, now the interesting part. Let's say you own that apartment. Well, you don't actually own the apartment. You own the number of shares represented by the apartment. You own those shares as a portion of the corporation. And, critical point, what goes with owning those shares is whats called a "Proprietary lease". Ownership of the shares entitles you to a lease on that apartment. Naturally, there are terms to the lease. I mean, a lease is a lease. You're "renting" the apartment, so to speak, as an entitlement for owning those shares. But you still have to abide by the terms of the lease, regardless. And, those proprietary leases can be very restrictive.
Now, the corporation is like every other corporation. The shareholders elect a board of directors; one share, one vote. The board of directors manage the finances of the coop; they collect the monthly payments from the shareholders (called "maintenance fee"

Of course, the board of directors is also responsible for all the other details of managing the building, such that the shareholders find their experience there suitable. If they don't like the way things are managed, they elect other directors.
The board and/or the shareholders can also change the proprietary Lease, but that takes a big effort. Two thirds of all shares must be voted for the change (not just 2/3 of whoever happens to be present).
An interesting detail of coop structure, which may or may not be relevant here, is that a committee of the board of directors decides who may and who may not buy shares in the corporation. Even though the seller and the buyer are happily settling between themselves a great deal; if the committee doesn't like the buyer, the deal is off. This is why, in general, banks don't like coops much, because of the extra level of difficulty involved in selling them. But in New York (well, Manhattan anyway), most owner occupied apartments are coops.
That covers the basics of the whole idea. It may or may not help as we determine the contract contents for "ownership" in Neualtenburg. Any other questions come up about this arrangement, please bring it up, and I'll try to answer.
Sudane