Today I applied for unemployment insurance. New York has an online application process. It wasn't too bad. It appears I will also have to claim for each week (Monday-Sunday) of unemployment within a one week window (Sunday-Friday) in order to collect that week's benefit. No benefit is paid for the first week of unemployment. After that the benefit is normally about half your previous wage (up to a maximum of $405/week) for a maximum of 26 weeks during the following year. Apparently the federal stimulus legislation has increased the benefit by $25/week and 33 weeks has been added to the 26 weeks for which you can collect.
I didn't sign up for direct deposit but will do so as it appears more convenient than the direct payment card (like a debit card) that you get otherwise.
Here is the New York unemployment insurance website.