There are many excellent schools uptown, downtown in the bronx and in brooklyn. Depending on age and location (not including nursery schools however unless they extend beyond nursery..) here is a short list of most sought after schools..
Upper East Side:
Trinity (coed, perhaps most academic overall)
Brearley (all girls, was traditionally considered to be more intellectual/academic of the 3 top all girls)
Chapin (all girls, more waspy and traditional of the 3 top girls)
Spence (all girls, something in between B and C)
Dalton (coed a bit more 'progressive' without being explicitly 'progressive')
Upper West Side:
Ethical Culture
Bank Street (excellent progressive school, nursery - grade 8 only)
Bilingual:
Lycée Francais
United Nations School
Riverdale Schools (Bronx):
Horace Mann - coed very very academic
Fieldston - upper school of Ethical Culture, more progressive
Downtown:
Friends Seminary (solid academics, very sought after downtown school)
Grace Church (traditional, sort of like an uptown school downtown)
City & Country (oldest/best progressive school in the country, nursery - grade 8)
Brooklyn:
Packer Collegiate (traditional, academic)
Brooklyn Friends
St Ann's (very high achieving top progressive education, very artsy etc like a much more academic King Alfred's.. they don't have grades but children regularly accepted to Ivy League schools etc)
Berekely Carroll (very well rounded, strong humanities)