Housing cooperatives, or co-ops, are common in New York City. John Lennon lived in one, and other famous people have wanted to join co-ops but are not always accepted. Members of a co-op can accept or reject the membership of newcomers. Living in this type of housing means that you buy the right to live there, like buying stocks in a corporation, but the actual owner is the corporation. Each member of the corporation, your other neighbors in the building, has the same rights. The agreement you sign when you acquire the right to live there is called a proprietary lease or occupancy agreement. It is not an actual real estate transaction, because the property is owned by the corporation.
Co-ops are similar to condominiums in the way they operate. They are multiple housing units that have a set of bylaws and elected officers. There is a monthly fee to cover maintenance and utility costs. The difference is in the ownership. The housing cooperative is a shared enterprise with one asset, the property itself. Selling your co-op means you are selling your stock (your housing unit) in the corporation. In some cases, you keep the profits from the sale. In other cases, the housing unit is sold back to the corporation at the original purchase price, and the profits are equally distributed between all the shareholders when the housing unit is resold.
The history of co-ops is part of the history of New York City. In 1876 the first housing cooperative was established on West 18th Street. It was called a "Home Club" and it was essentially a way for wealthy people to own a home without the responsibilities of home ownership.