Favorite places to eat in Manhattan?

You might want to book some reservations as many of the places people mentioned can be hard to get in to. My favorite place is Union Square Cafe. they do brunch / lunch and dinner. I like Co. for pizza or John's of Bleecker st (greenwich village) Lucien (east village) for mussels frites or other classic French bistro dishes. Maison Keyser (multiple locations) is nice for breakfast /lunch, great croissants. Locanda Verde (tribeca) . Some many choices.... Enjoy your trip.

Totally forgot, do not miss Breads Bakery!

SO. MANY. PLACES! Here's some of my favorites:
Xi'an Famous Foods (several locations throughout Manhattan) for great authentic Chinese food
Black Seed Bagels
Russ and Daughters in the LES for bagels & lox (you can also try Sadelle's, but honestly I think R&D is better and the wait is not as bad)
Cosme or Eleven Madison park for a high-end dinner
Dizengoff NYC inside of Chelsea Market (and also don't miss Seed & Mill while you walk by Chelsea Market, their halvah goat milk ice cream is to die for)
The Spotted Pig
Le Relais de Venise L'Entrecote in Midtown for wonderful steak frites
Katz's Deli for a Jewish deli

And if for some reason you find yourself in Brooklyn (which I hope you do, because the food is fantastic as well), you should try Pok Pok Ny in Brooklyn Heights (authentic Thai food) and Dinosaur BBQ in Park Slope. Also Talde in park Slope. Gosh, so many places to go!

It really depends upon what you are looking for in terms of cuisine, neighborhood, price point. There are so many great places to eat in New York. And if you include the boroughs, the list would be endless.
My favorite Sunday brunch or Mexican at any time is Fonda on Avenue B between 2nd & 3rd Streets. Check out their web site. I don't like their other locations as well.
Kayser Bakery on Broadway has great bread as well as a nice lunch.
Cafe Sabarsky, Tocqueville, Botequim, Blue Smoke/Jazz Standard, Minton's, DB Bistro = again, it really depends upon what you're in the mood for and what part of town you want to be.

There are so many places! On my last trip to NY, last summer for the Fancy Food Show, I ate at the Breslin twice (dinner and breakfast). I stayed in a hotel in Koreatown, and the Breslin is just a couple blocks away. I also had a lovely multi-course meal at the Korean Gaonnuri. In the past couple of years I've also enjoyed meals at Bobby Flay's Gato, Empellon Taqueria, Fatty Crab, Hakata Tonton, Ivan Ramen, Kung Fu Little Steamed Buns Ramen, Mission Cantina, Saki Bar Hagi, and Xi'an Famous Foods. Have been to the Modern and Le Bernardin as well, but those are more in the "most famous" places to try. You might want to check out New York Magazine's various dining guides online for suggestions. I like to plot out restaurants on a Google Map, giving myself lots of options depending on what neighborhood I end up in.

I'm a semi-regular visitor to NYC, and here are a few of my favorites:
Cosme (exquisite Mexican food) - I have only been here once (this winter) but it was one of my very favorite meals ever.
City Bakery/Birdbath Bakery - For me, City Bakery exists exclusively for the pretzel croissant. Birdbath is an offshoot of the original City Bakery and has multiple locations throughout the city (which all have the pretzel croissant as far as I know). It is probably good I don't live closer to this.
Soba-ya - Soba noodle focused restaurant, with noodles made in house.
Amor y Amargo - Excellent and unusual cocktails focusing on bitters and amaro (only stirred drinks). The space is tiny and busy but always seems to have a little corner available, even when other cocktail bars in the area have bouncers in front with 2 hour wait lists (A & A doesn't do this).
Mayahuel - Complex tequila and mezcal-focused drinks (they also serve some food). This place does have the bouncer/2 h wait situation on weekend nights, which you can avoid by going early (or maybe they take reservations?).
Not Manhattan, but Smorgasburg is fun and you can take the East River ferry to it from Manhattan which I also like.

Here's a past thread on the topic that might be a good place to start: https://food52.com/hotline/20690-nyc-experts-looking-for-favorite-affordable-casual-restaurants