I lived and dined in The City for 8 years; dining out an average of 1.5 meals a day -- and I don't eat breakfast. I lived in Brooklyn Heights and ate in Brooklyn quite a bit -- most of the places I thought were the best were smaller "holes in the wall", but I'll try to give you coordinates.
Steak -
First off, no offense intended to other posters, but avoid Sparks. I've eaten in every "known" steakhouse in NYC at least a couple of times and on the island of Manhattan, MHO is that The Post House is the place, hands down. It's at 63rd street between Madison and Park Ave (nearer Park). The Post House is also a great place to take non steak-fans, as their menu includes other fare that is very very good.
Keens is also good (and very cool bar/fireplace) with great chops. It's on 36th between 5th and 6th (nearer 6th). So that's your midtown place to go. Gallagher's is an old-school steakhouse near the theater district, on 52nd between Broadway and 8th. I know John Cramer is partial to Ben Benson's, and I like it, but if you're only going to go to one steakhouse in NYC the aforementioned are better.
HOWEVER, if you'll be in Brooklyn anyway, and you're at all a steakophile, you MUST go to
Peter Luger's. Many believe, including your humble narrator, that it is the best steakhouse in the city if not the country. Order a slice of bacon per person (it's not on the menu). and BRING CASH. They take only greenbacks or the Luger Credit Card (one of which I possess). If you like steak and miss Luger's you've done yourself a disservice.
Gage & Tollner is in downtown Brooklyn and it's good, but if you're in Brooklyn anyway, go to Luger's.
Italian --
Avoid Little Italy. There are some pretty good places there, but if you want the best Italian food to be had, go to Volare (West 4th and 6th Avenue on the East side of 6th and the North side of 4th). You have to go downstairs to get to it and there'll be only a red awning to tell you you're there. I am not kidding about this place -- it is a gem.
Cuban -
Victor's -- right next to Gallaghers (see above). Get the "Fried Cow" if you're in a meat mood, and as many fried plantains as you can stuff in your pie hole.
Wings/sports -
Scruffy Duffy's. 8th Ave between 46th and 47th. No, I do not own stock in this place ... but it has the best wings in the city and is a great sports bar. I was there 3-4 times a week for 8 years. Mention JP to the bartender if you go. Pat Hughes is the owner and if he's there, give him my regards. There are likely to be as many off-the-boat Irish in that place as there are Americans. The hot blonde is Triona, and she's taken.
Foofy Dining -
One if by Land is incredible dining, and I'm not a fan of foofy dining. It costs slightly less than open heart surgery, but it's worth it. It's just off of 7th Avenue; and I find it by Zen -- but any Zagat will have the coordinates. I'm sure many of our fellow pelicans will have foofy dining suggestions, so I'll leave the remainder of this category to their expertise.
Seafood/surf-turf
City Hall on Duane near Broadway. Great seafood and beautiful place. I'm a pan fry kind of guy, and they do 'em right.
Burger -
Best burger in the city, hands down, is at Walker's on Varick and North Moore (in TriBeCa). Great ginmill with very good food and an exceptional burger.
Sushi -
In Manhattan, avoid Nobu; it's not worth it. Find Blue Ribbon Sushi -- it's in the West Village at Sullivan and Prince. Best damned sushi I've ever had (including in Japan).
In Brooklyn, there's a place on Clark Street, up in Brooklyn Heights called Iron something (Horse?). A very close 2nd to BRS IMHO and cozy.
Mexican -
Maya. No doubt. On 1st Ave between 64th and 65th. It's what Mexican food is meant to be. It's also not cheap.
Asian -
Asia de Cuba -- for lunch. It's mid-town and announced only by a small sign. Madison between 37th and 38th.
Rain -- very good w/ multiple locations. This was my default Asian food dining choice for years.
Lucky Cheng's 1st Ave between 1st and 2nd.
I can't tell you about this place if you haven't been there, but go anyway. The food is good, but the experience can be life-changing.
Moroccan -
It is well worth it to go to Chez Es Saada on 1st Street between 1st and 2nd Ave.
Deli -
Katz's Deli on Houston. Tip the guys (there are plates on the counter). Your succulent pastrami sandwich will magically grow for a buck or two.
Chinese-
Many people swear by Wo-Hop. I like Wo-Hop, even though EVERY time I've been there I've seen cockroaches. It's on Mott Street south of Canal in Chinatown.
However, I prefer Lin's Dumpling House, which is just up around the corner from Wo-Hop.
If you're in Chinatown, you must go to the Chinatown Ice Cream Factory. MMMmmmmm Green Tea Ice Cream....
Indian -
Cafe Spice. Yummmfreakingeee. Don't go to the one in basement of GCT, go to the mothership at University place between 10th and 11th. Cannot go wrong. I miss this place.
Fun/Eclectic Dining
Ruby Foo's is fun and a bit hectic w/ good food.
At the end of an evening, have your cabby take you to Veselka -- a Ukrainian diner open 24/7. Great food, great coffee. 2nd Ave and 9th St.
EDIT - forgot Pastis, which is in the meatpacking district at 9th and Little West 12th. A bit trendy, and wayyyy too Fronch for my usual taste, but good steak frites and other food. OK, I've only ever been there inebriated, but it merits a mention.
Brooklyn Heights -
On Henry Street (up near the Brooklyn landfall of the Brooklyn Bridge) are two restaurants definitely worth the visit -- one is Henry's End, to which I brought everybody I dated and every friend that visited. It is phenomenal food, with game options year round. It also has a great and reasonably priced wine list.
Just next to Henry's End is Noodle Pudding, which is another fave.
Cobble Hill --
Literally around the corner from my former residence is a small, incredible place called Cafe on Clinton. Strangely enough, it's on Clinton Street, just about where it intersects Warren. You can't have a bad meal there, and they have a signature dish that is excellent.
Smith Street in Cobble Hill area is infested with new (last 8 years) restaurants that sprung up as the area yuppified. Ask the locals which of these are the best currently. You can't go wrong with The Grocery. And have a drink at "Boat" after you're done. Play "Marquis Moon" by Television on their jukebox, too.
That's about 2% of the options I'd recommend. If you have a specific thing in mind, let me know. After clients came into town and I'd taken them out to dinner, their partners/friends/family would call me for restaurant recommendations when they came into town. My info is about 2 years old, but I'm talking about the classics that shouldn't ever change.
Good luck and good eats.
JP