If you're familiar with HOAs, a co-op is basically an HOA for an apartment building.

Technically you don't own the unit you live in, you buy shares in the co-op which entitle you to live in the unit you live in, but there's not much of a practical difference.

The practical issues come in with respect to how you use the apartment. Co-op rules usually prohibit subletting and require getting approval before doing renovations (and also time restrictions on renovations) and approval to eventually sell your unit down the road.

Generally speaking, they aren't terrible. Unlike HOA horror stories, co-ops usually don't care about superficial stuff like how you decorate your unit. The rules are primarily there to prevent you from negatively affecting the units around you (i.e., you can't hire some fly-by-night plumber to smash a hole in the wall and install a jacuzzi that ends up flooding the unit below you).

But maybe this is stockholm syndrome talking because there's almost no alternative to co-ops in NYC.

Some units are condos where you do actually own the unit. And that generally comes with a higher purchase price, fewer restrictions, but still some use restrictions because, again, you can't do stuff that would damage the units around you.

The last alternative is buying a house but it sounds like that's not a good fit for you given that you're considering studios.