The East Village Tavern at East 10th and Avenue C is an unknown, unpretentious gem. The food is good and they have an absurdly large number of beers on tap -- usually including at least one cask "real" ale. They sell "flights" of beer which are a great way to try a ton of beers without getting completely s-face.
Down the street head for Zum Schneider if you want to have a real Austrian/Bavarian experience. Unlike the East Village Tavern, it gets packed.
Mission Dolores at the corner of 4th Ave and Carroll St in Brooklyn. It's pretty small: enter through the wind tunnel, across the little courtyard and then in. Excellent rotating selection of craft beers at GREAT prices ($5-$7). Great people to meet, pinball, dogs allowed. No food, but there are a few places close by that will deliver. Take the 'R' to Union St and walk two blocks south.
As for the city? A long time favorite that's been around since Moses wore short pants: The Holiday Cocktail Lounge, St Marks between 1st and 2nd Ave. Eclectic crowd, good prices, good jukebox.
I love the Double Windsor on Prospect Park West near 16th St. They have a great selection, and you can buy a flight. It's a laid back Brooklyn bar, there is food, but it's separate and I think it's all cash only as well.
I visited New York City a few weeks back and thought McSourley's was really cool. I like the history behind the place, and the fact that its a presidential hangout spot; kinda neat. I also liked Flight 151 in Chelsea right by my hostel. Pretty quiet little sports bar.
McSourleys back in the 70's and 80's was great - still good but not like it used to be. Was a lot more friendlier back then - working class and college crowd