Condo
In a condo, you own your physical apartment, represented with a deed from the State of New York. However, you’re also required to make monthly payments to maintain the “common” parts of the building. You participate in electing a board to represent the owners that oversee the maintenance of common areas (elevators, hallways, heating equipment, etc.). The board may require an application for review but typically cannot reject an applicant unless they themselves buy the apartment through a right of first refusal document.
If you’re thinking of renting out your apartment in the future, a condo would be a better choice than a co-op.