How do ownership housing co-ops work?

Non-ownership housing co-ops are simple enough to imagine: basically you're renting the house, but instead of paying a for-profit land lord, you pay a nonprofit designed to benefit everyone who lives in the apartment complex. But how would ownership housing co-ops work? Does the initial price of buying the house go to the co-op and then they collect monthly fees from you and anyone you sell to like a homeowner's association? Are there mixed ownership and non-ownership housing co-ops where you can either rent or own?

Finally, do you personally prefer housing co-ops to be ownership or non-ownership?