Cooperative apartments (“coops”) differ from condominiums (“condos”) in several ways. When you buy a coop, you buy stock in the corporation that owns the apartment building. The building then “leases” the coop to the buyer under a long-term proprietary lease. Coop owners pay monthly maintenance to the building corporation for items such as the expenses of maintaining and operating the building property, property taxes and the underlying mortgage on the building (if any).
http://www.nyccounsel.com/real-estate-condos-coops/what-is-t...

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A housing cooperative, or co-op, is a legal entity, usually a corporation, which owns real estate, consisting of one or more residential buildings; it is one type of housing tenure. Housing cooperatives are a distinctive form of home ownership that have many characteristics that differ from other residential arrangements such as single family home ownership, condominiums and renting.[1]
The corporation is membership-based, with membership granted by way of a share purchase in the cooperative. Each shareholder in the legal entity is granted the right to occupy one housing unit. A primary advantage of the housing cooperative is the pooling of the members’ resources so that their buying power is leveraged, thus lowering the cost per member in all the services and products associated with home ownership.
https://en.wikipedia.org/wiki/Housing_cooperative