food unique to New York
Katz's for Pastrami
Sushi Yasuda for nigiri (pricy but changes your
perception of nigiri)
H&H bagels with whitefish
Pizza. I like Lombardi's even though they've slipped since the renovation. Try it with the pancetta and roasted peppers.
Try Lupa or better yet, Babbo for pasta and see how it contrasts with Oliveto and Quince.
Pastis for steak frites.
Otto for grilled pizza, cheese, and great wines.
And if Jean-George tasting if you're looking for a blow out dinner.
Don't miss out on Veniero's on 9th and 1st Avenue.
There are other places where I like specific Italian desserts, but since you narrowed it down to E Village, you will not find anywhere in the Bay Area anything close to to the Italian cookies, cannolis and coffees at Venieros's.
What comes to my mind are places that have 'history' and are not the bistro of the week. With that said, here's my list. I'm sure I'll think of others later. It's always like that.
Katz's - same atmosphere since the beginning of the 20th Century. The customer takes a check from a turnstile as they enter just like the old days and you'd better not lose that check or you're going to pay even if you only had a hot dog.. As other hounds have stated, they have the best pastrami.
Yonah Schimmel's for knishes - not far from Katz's. Like a time warp and a great assortment of knishes.
Barney Greengrass - check out the art deco refrigerator cabinets from the 1930s. Not much has changed and the bagels and smoked salmon are wonderful.
There's a little Jewish bakery on Second Avenue around 6th Street (which name escapes me) but they've been around for about 100 years. I love the poppyseed strudel and the atmosphere. I don't remember if there are any tables but the challah and pastries are great.
Zabar's - totally unique and New York. There's no place to sit except a separate dessert and coffee space that's attached - but you need to visit them at least once and try some samples of cheese and other delicacies. They've also been around for years.
Sammy's Romanian Steakhouse on Christie Street. Another landmark that's totally New York. Go with a big appetite.
Keen's Steak House - olde New York and probably has landmark status.
flynn, the "little jewish bakery on second avenue around 6th street" is none other than moishe's, right?
to the original poster: why limit yourself to the east village? lots of great, unique food options throughout all five boroughs and other areas of manhattan, though i realize that as a tourist you probably have only a limited amount of time to work with.
some classic ny eats include pizza, italian, falafels, egg creams, cheesecakes, italian ices, bagels and lox, and pastrami on rye.
agree with everyone above about katz's and barney greengrass for the food, atmosphere, and history; both are definite must-dos. if you don't want to stray too far from the east village, get bagels at murray's on 6th ave - not the absolute best in the city, but easily one of the best downtown. and get bialys at kossar's on grand st on the lower east side - also close to the EV. on the same block as kossar's is doughnut plant - opinions about this place are very split on this board but those who love DP's donuts really swear by them.
as for pizza, i think you should really make one exception to your plan to eat only in lower manhattan and take the Q train to midwood, brooklyn for pizza at di fara. yes, a place like lombardi's is close to the EV and most out-of-towners would probably think it's great, but there's nothing even close to the level of di fara in this city, so you should really consider it.
and if you're adamant about not leaving downtown, go to arturo's on houston street in the village. i like it better than the other famous downtown pizza places (john's, lombardi's, joe's), and it has great old-school ny atmosphere to boot.
lots of great italian options downtown. if you want to do super upscale/nouveau, try to get a reservation at babbo. something a bit more casual but with superb food nonetheless is lupa on thompson st. piccolo angolo is a reasonably priced and homey (albeit loud and cramped) place on hudson st with good northern italian food. and there are tons of cheap pasta houses in the east village as well (like max), but i'd recommend a mid-priced, somewhat less casual option nearby like bianca over those.
for falafels, go to taim on waverly place or azuri cafe on w. 51st st.
for egg creams, go to the aforementioned gem spa on 2nd ave/st. marks place or the unnamed place on the west side on avenue a between st. marks place & e. 7th st (the sign outside simply says "belgian fries"--longtime locals refer to the place as "ray's candy store", though).
two of the best italian ices in the city are benfaremo - the lemon ice king of corona and ralph's. both are based in the outer boroughs but i think their ices are available at various shops/pizzerias around manhattan.
for cheesecakes, lots of people swear by junior's in brooklyn, but i haven't been to their newish times square location, so i can't vouch for the qualiity there. closer to the EV is eileen's special cheesecakes on cleveland place in nolita. and one of my current favorites is at the french brasserie artisanal (on the corner of e. 32nd st & park ave), which is not made in the traditional ny style but is delicious nonetheless.
though i think the food is generally just so-so, you should definitely check out one of the old-school ukrainian diners on 2nd ave (such as veselka) in the EV for pieorgies, blintzes, kielbasa, and challah bread. 1 1/2 blocks south of veselka, on the other side of 2nd ave, is b & h dairy luncheonette, which is a great old-school jewish dairy luncheonette with very good pierogies and soups.
blue ribbon in soho for really great food, including both standard and unique eclectic dishes. it's easily one of the best downtown, late-night, upscale dining options.
also, lots of great food carts around the city. sam the red chef serves kimchi hot dogs and bulgogi burgers on the LES, and lamb or chicken over rice are served on the sw corner of w. 53rd st & 6th ave in midtown.
once the weather gets warm again, shake shack in madison square park for burgers, hot dogs, and custard shakes.
and finally, i think you should get brunch at prune in the EV, which is a really great all-around restaurant with some unique dishes, even at brunch. note that lines are super long here at all times. other downtown brunch options include extra virgin, august, five points, and clinton street baking company.
My friends from Cali love falafel. We're always forced to go for falafel and pommes frites whenever they're in the city. And that's all in the East Village.
You could also try Stanton Social for a very New York scene. If you did a bit of travelling: Momofuku, Zum Schneider, Sunburnt Cow, 7B, you could traverse the entire globe in a couple hours and within just a few blocks. That's a total New York experience.
No-one has mentioned the Oyster Bar in Grand Central Station! This is one of my all-time fav New York joints. Chowder at the counter is fantastic. Oh, and do bother sitting at a table. Go to either the high counter, where they shuck the oysters and prepared the chowders, or the low, u-shaped counters, presided over by old-style New York waitresses, all of them called Maureen.