I've not checked books out of the New York library, but I have used my sister's library card to check books out of the Seattle library. Since both libraries use the same technology, I'd expect the New York library to be pretty similar to Seattle's library in regards to checking out eBooks.
Library eBooks are usually available in 2 formats: Adobe Digital Editions (PDF) and Mobipocket format. The books are DRM protected, so you will need to either activate your device through Adobe's DRM server or register your device's PID with the library to read Mobipocket's DRM books.
I can check out the Mobipocket books and read them on my Cybook Gen3 ebook reader because I can activate that device through Mobipocket's DRM server, but I cannot read Adobe Digital Editions books on my Cybook Gen3 because that device does not support DRMed PDF files and cannot be activated through Adobe's DRM server.
All I need to check books out is my sister's login and password. When I check books out, I download them to my computer and them copy them over to the Cybook Gen3.
The eBooks time out after 21 days. So after 21 days, I can no longer access the book. If I want to keep the book longer than 21 days, I have to check the book out again after the book has timed out. With Adobe PDF books, you can "return" a book, which means you free up the book for other folks to check out before the 21 day period of time, but it isn't necessary to turn the book back in (I have read Adobe PDF library books on my computer).
The Sony Reader PRS-505 (or newer) with v1.1 firmware (or newer) supports DRM activation for Adobe PDF, but does not support Mobipocket formated books. Some books come only in PDF format; some books come only in Mobipocket format and some come in both formats. When you review the online book catalog you'll see what formats are available. You can review this link to learn more about this:
New York Public Library FAQs about Adobe PDF books