I asked this question of a friend of mine and they sent me somewhere I never would have found on my own. Her directions:
"The bar with no name, down some seedy looking stairs on Rivington past Suffolk. It's nice inside - very, very dark, and has interesting people."
I figured she'd been too drunk to see the sign, but it really has no name and you will think twice upon seeing the entrance. Needed our passports to get in, like everywhere. Was a great find though, I'll be back there again.
Then we went on to find a load more good bars in the Lower East Side.
One of the best ones was Iggy's Keltic Lounge, 132 Ludlow St. This is an Irish Punk Pirate bar, and it was a lot of fun. Walked in and they played the Clash and Rancid within the first 5 minutes, so figured we'd stay for a bit. Scruffy as hell, toilets covered in graffiti (yet still clean) and a great friendly place to drink. (There's another Iggy's uptown but unconnected I believe).
We asked Iggy where we could go next, he sent us down the road to a Detroit themed bar (everything made out of cars, walls all black, music depressing). We didn't like that one much.
Best of all was Niagara bar
http://www.niagarabar.com/contact.html - this is the one on the corner of Tompkins Square Park on Avenue A. You'll recognise it by the large Joe Strummer mural on the wall. It has great atmosphere, a wonderful selection of beers and single malts and best of all there is a downstairs 'Tiki lounge' where you can go dancing on a Friday and Saturday night. They play the Ramones (of course) and a whole mix of stuff. Then there's some more good bars on the way back up Avenue A to 1st Avenue Station.
Have a great time!