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.
The story was about the New York City Board of Elections, which must choose between two systems that were recently certified by use by NY State Board of Elections. Proposed systems by other vendors did not complete the certification process.
Voters will mark their ballots by hand, and then insert them in to a scanner/counter which will check for errors (overvotes, etc) so that the voter may decide to make corrections before the ballot is accepted and counted.
There are also vote marking devices, which provide for various types of interfaces such as touch screens, and that then mark a ballot. In some cases the marked ballot is then placed into the precinct scanner, and in others it is directly deposited into a ballot box.
The paper trail is the the pre-printed ballot with the choices marked on it.