This is the shortest version I could find of what takes place at http://www.nysccc.org/Legislative%20Information/howbill.htm
The Legislature convenes on the first Wednesday after the first Monday in January. Early in the session the Legislature meets on Mondays and Tuesdays. Later more meetings are held.
Bills are introduced and each is given an introductory number which identifies it throughout its progress. Each time the bill is amended, a letter is added to the number, e.g. 21A, 21B, etc. After introduction, the bill is referred to a standing committee.
Committee action is probably the most crucial step in the legislative process. Except under certain conditions, no bill is taken up for final consideration until it is reported out of its committee. An affirmative vote of a majority of all members of a committee is necessary for a report on the bill. The committee may report a bill favorably, unfavorably, or merely for the consideration of the appropriate house. Many bills are not acted on at all.
Passage of bills - A bill must be on the desk of members in final printed form for 3 days before it is voted on, unless the Governor issues a message of necessity. To pass, a bill must be voted favorably by a majority of those elected to each house. When a bill is passed in one house, it is sent to the other. If a companion bill has been introduced in the house receiving the bill, the passed bill may be substituted for its companion in the same order of business. When a bill passes both houses, it is sent to the Governor.
Action by the Governor - During the session, the Governor has 10 days, exclusive of Sundays, to approve or veto a bill. If no action is taken within that time, the bill becomes a law. If vetoed, it may only become law if repassed by a 2/3 majority of each house. After the session adjourns, the Governor has 30 days within which to act. Bills not acted upon within that time are dead. In practice, the Governor does not practice the "pocket veto." He either signs or vetoes all bills sent to him. Bills passed during the last 10 days of the session are considered 30 day bills.
Since the great majority of bills reach the Governor during the 30 day period, he and his staff are faced with a great concentration of work during this short period. If you wish to communicate with the Governor at this time, it would be most helpful to keep your remarks brief and to the point. The correct subject, number, and the introducer of the bill should be included.