If your country of citizenship is eligible for the Visa Waiver Program [1] you just need to fill a form [2] online, declaring you're not a terrorist, drug dealer, child abuser/abductor, etc. [3]

It is probably a good idea to get some health insurance for your whole stay, unless your country has some kind of arrangement with the U.S. to this effect (and I don't know if any country does). You can get insurance from your travel agency or separately online. Check the terms of your home insurance or credit card, if any. Some include some sort of travel insurance.

[1] http://travel.state.gov/visa/temp/without/without_1990.html

[2] https://esta.cbp.dhs.gov/esta/

[3] Seriously!

You should be fine assuming you can legally be in New York for the entirety of the batch (12 weeks). We've had several non-US citizens. We're not an official school, so we can't sponsor student visas.