The only good nazi is a dead nazi. If I was one of these folks I'd be more concerned about someone following me home after their silly clown parade is done.

After the NYC bed bug nightmares of the past few decades I'd assume offloading a used mattress on anyone in NYC is an impossible feat.

While everyone else's explanations are pretty adequate at explaining what a co-op is one thing that everyone left out is the crazy maintenance fees, these tend to go up every year. I've got friends who have paid off their units but their monthly maintenance fees are more than what some people pay in rent. Every co-op board is different but they're also a lot harder to sell should you choose to move out. Some co-op boards can also make your life difficult. YMMV.