On December 15, the New York State Board of Elections said that the state will soon buy optical-scan vote-counting machines. Still to be decided is whether Election Systems & Softward (ES&S) of Omaha, or Dominion Voting Systems of Toronto, will get the contracts. See this TV news story, concerning the process in New York city.
Voters will electronically mark their paper ballots, which are then read by the vote-counting equipment. There is a paper trail. New York in 2008 was the only state still using mechanical voting machines, which were invented in 1892 and provide that voters pull levers to indicate which candidate is being voted for. There is no paper trail with mechanical voting machines.