Best resources for finding an apartment in Lower East Side, New York City?

Start off on Craigslist for the "no fee" and "by owner" sections to try to avoid a broker fee. You may just get lucky. If that doesn't work (which is often the case) you'll probably have to go through a broker. They list predominantly on Craigslist or you can go directly to sites for brokers such as Bond and Citihabitats and see what they have listed. A broker fee is about 10% of a year's rent, so just be warned that it's pretty pricey. But, for the most part, they're unavoidable and they will find you a place in about 1-3 days.