Can my fiance move into my apartment with me, even though I am the only tenant listed?

Your right to share your apartment with another person is governed by NY Real Property Law §235-f, which makes certain restrictions on occupancy unlawful. This law provides that:

"... 2.  It shall be unlawful for a landlord to restrict occupancy of residential premises, by express lease terms or otherwise, to a tenant or tenants or to such tenants and immediate family. Any such restriction in a lease or rental agreement entered into or renewed before or after the effective date of this section shall be unenforceable as against public policy.
3. Any lease or rental agreement for residential premises entered into by one tenant shall be construed to permit occupancy by the tenant, immediate family of the tenant, one additional occupant, and dependent children of the occupant provided that the tenant or the tenant's spouse occupies the premises as his primary residence..."

Your fiancee is considered an additional occupant. As long as her additional occupancy does not constitute a violation of federal, state or local  laws, regulations, ordinances or codes, then the landlord cannot restrict your right to occupy the apartment with her. This law does require you, as the tenant, to occupy the apartment as your primary residence.

§235-f (5) requires the tenant "...to inform the landlord of the name of any occupant within thirty days following  the commencement of occupancy by such person or within thirty days following a request by the landlord."

Lastly, §235-f (9) does give you specific rights should the landlord violate this law:

"Any person aggrieved by a violation of this section may maintain an action in any court of competent jurisdiction for:
    (a) an injunction to enjoin and restrain such unlawful practice;
    (b) actual damages sustained as a result of such unlawful practice; and
    (c) court costs."