New York System is a Rhode Island term.  Pretty much the same as Coney Island Dogs or Texas Wieners.  A very small dog, only four inches long, grilled whole and topped with "meat sauce," mustard, onions and celery salt.

Unless you were born in Rhode Island they are an acquired taste.  The sauce is strange.  I figure ground beef, chili powder and meat.  Maybe a few other spices.  Nothing like Coney Island sauce elsewhere.  But by the time you've started on your second, third or fourth dog, you'll start liking them.  Washing those "gaggers" (Rhode Island for hot dog) down with a glass of coffee milk seems to help.

Watch the guy at the grill build the dogs.  Lines 'em up on his arm and piles on the onions, sauce, mustard and celery salt.