Things you can only do in Brooklyn?

Along with the Brooklyn Heights Association Inc as Jaclyn mentioned, spend an afternoon walking around DUMBO. Juliana's Pizza, Jacques Torres Chocolate, and Jane's Carousel are all things worth checking out.

Fort Greene is a great neighborhood, though I'm certainly biased on that account. Brooklyn Flea Market has a location nearby, depending on the date - check out their website for more info. Walk around the area south and east of Fort Greene Park, then head downtown. Try the cheesecake at Junior's and walk along Fulton Street Mall to get a feel for the area. Continue on to Cobble Hill, another cute neighborhood, if you feel like it.

The Brooklyn Museum and Brooklyn Botanic Garden are also worth seeing, especially if you're there at the right time for the cherry blossoms. Nearby Prospect Park is beautiful as well, having been designed by the same planner as Central Park.

Finally, Coney Island (New York) is definitely an "only in Brooklyn" experience - go if the weather is nice and stop by Nathan's Famous for a hotdog. Luna Park and the famous Cyclone roller coaster open at the end of March, but to be honest, Coney Island is more about what it once was than any of its current attractions.

For the best pizza you'll ever eat, you MUST visit L&B Spumoni Gardens!  Then head down to the boardwalk on Coney Island (New York) and grab some ice-cream at Coneys Cones.  Enjoy your time here!

Brooklyn Bridge and  Brooklyn Heights Promenade. A walk around Williamsburg for lunch and some shopping. A performance at Brooklyn Academy of Music.

My favorite part of Brooklyn is just roaming the streets around Park Slope neighborhood. Walk up and down 5th Avenue; the area around 1st street and 5th is the top neighborhood in the NYC area to live! Beautiful shops and restaurants. Also, quite a few celebrities live above 6th Ave near 3rd and 4th streets, including Maggie Gylenhall and Gary Oldman.

2 things: Pizza and Banya

For the best pizza of your life - Roberta's, Di Fara Pizza, Franny's, or Paulie Gee's

To escape the cold and spend the day sweating it out while feeling like you stepped into another country: Russian Bath on Neck Road or Sandoony USA

Talia,

Well this might seem obvious but you have to visit the Brooklyn Bridge There is also a great restaurant to eat at called The River Café located right under the bridge and has great views of NYC especially at sunset. Its an expensive restaurant but the food and atmosphere make it totally worth it.

Go to the Wythe Hotelrooftop for an amazing view of Manhattan. It's an enclosed bar with the option to be outside if you want to brave the cold. 

If you like Deep House, check out Output or Verboten.

Free comedy shows. Dumbo Comedy is the first Tuesday of every month at DUMBO KITCHEN Brooklyn Bridge Park Basketball Courts has amazing views of Manhattan. Go to Juliana's Pizza for the best pizza and Ample Hills Creamery for ice cream. Walk across the Manhattan Bridge for views of the Brooklyn Bridge and then walk the other one on your way back. 

Polly's answer below is right on--Coney Island (New York), Brooklyn Flea Market, Prospect Park West, Brooklyn Museum, etc...

McCarren Park!

I haven't been there yet, but I'm dying to see the The City Reliquary. It's a museum of New York-related memorabilia that's supposed to be really fun. 

Prospect Park  is my new favorite obsession. It's basically the Central Park of Brooklyn (I think it was even designed by Olmsted, too). So, so beautiful and much less crowded than Central Park. 

The Brooklyn Public Library at Grand Army Plaza is one of the most beautiful buildings I've ever seen in my life. I would make sure to see this. 

Court St and Smith St are GREAT shopping/dining streets if you just want to walk around a bit. If you like lobster, Buttermilk Channel is great. 

My favorite Sunday activity is a stroll down the Brooklyn Heights Association Inc. While it's known more for its view of the city and the people watching, I love it for the brownstones right behind it--they're just about all covered by trees (you almost wouldn't even know they were there) but they're some of the most beautiful houses in the borough. 

I recommend the Brooklyn Museum