If you're visiting New York City (or even if you live here), these are the most popular free days that we should all be aware of.

Tuesdays

Brooklyn Botanic Garden, free from 8am - 6pm
The 9/11 Memorial Museum, free from 4:30pm - 7pm, and it's best to show up for this to get tickets at 4:15pm or so and they'll give you tickets with a time to come back at (which isn't a crazy request since it's free). If you're looking for things to do before your allotted time, there's a Century 21 across the street (not the real estate company, it's kind of a large department store) or you can see the Twin Towers water memorial infront of the museum.
Look at stars on The High Line - on Tuesdays through October, astronomers host free stargazing nights at New York City’s elevated park, where you can gaze up at the stars and planets via high-powered telescopes. Tuesdays through October. The High Line is between W. 13th St. and W. 14th St. at 10th Ave., 212-500-6035.

Wednesdays

Bronx Zoo, free from 10am - 6pm
The Staten Island Zoo, free from 2pm - 4:45pm, pretty sure the Staten Island ferry that you have to take to get there is free, too.
The New York Botanical Gardens, free from 10am - 6pm

Thursdays

The New Museum (on Bowery), pay $2 from 7pm - 9pm
Smaller Art Galleries in Chelsea, completely free. Sometimes has snacks.
The Museum of Arts and Design (Columbus Circle), free from 6pm - 9pm on Thursdays and Fridays

Fridays

The Museum of Arts and Design (Columbus Circle), free from 6pm - 9pm on Thursdays and Fridays
Museum of the Moving Image in Astoria (36-01 35 Avenue), free admission from 4pm - 8pm
Museum of Modern Art (MoMA), free admission from 4pm - 8pm
International Center of Photography, free from 5pm - 8pm
The Whitney Museum, free admission (or pay .25 cents, can't remember) from 6pm - 9pm
The Aquarium at Coney Island, Aquarium Admission on Summer Fridays goes from 4pm - 5pm (before Sept 2) and on Fall/Winter/Spring Fridays from 3:00pm - 3:30pm (after Sept 3) is by pay-what-you-wish donation.
The Noguchi Museum in Astoria is free on the first Friday of every month.

Saturdays

Sundays

Everyday

The Metropolitan Museum of Art, it says that their admission is $25, but if you look closer that's the recommended price. Don't be a sap and pay that. Every single time I go, I pay either .25 cents or $1 depending on what I have with me. Maybe that's being cheap, but they allow it so I say do it. All you do is say, "One ticket, please" and hand over what you want to pay. Also, they have a rooftop exhibit area that has a really nice view of the park. More thoughts here.
American Museum of Natural History, this place is the same deal at the Metropolitan (above), so you can pay what you want.
Sabon, if you ever see a Sabon store (it's a Body Shop type place), go inside and they'll ask you if you want to have your hands put through one of their routines. This isn't a thing only women do, they have a "gentleman" line and it makes your hands wicked and soft and great. If they don't ask you, then loom around the fancy sink they have in the middle of the store and then they'll ask. I tell people to get this done all the time 'cause it's great.
There are cheap movies days and here's a listing of them.
Real life night court. More on that here.
The arcade in Chinatown. It's small and not free, but cheap. Open all the days of the year. More here.
In the summer, the outdoor pool in Astoria. It's free as long as you bring your own lock.

Other Not Free Things, But Still Cheap

Renting row boats in Central Park.

Also this site (89th and Broke) is a really good site to bookmark 'cause of how many current/free things she blogs about.

Also, Bryant Park has tons of free stuff and sometimes free classes on things all the time (like juggling and beginner Italian for example).