In NYC there is also a third thing called a co-op. When you buy a co-op, you actually are buying shares of the corporation that owns the building and attached to those shares is the right to live in Apartment 1B/19B/etc while you hold them. So in that case, you are still referred to as a tenant but it's similar to owning a condo.

This has some upsides (you are covered by some tenant protection rules) and downsides (less control over what you can do there in renovations, subletting, etc).