The New York City Metro area has three different commuter rail operations
Long Island Railroad runs a mix of multiple unit electric and push-pull diesel as well as dual mode electric push pull diesel. The LIRR runs between New York (Manhattan, Brooklyn, Queens) ands Nassau and Suffolk Counties in New York State.
Metro North Rail Road runs push pull diesel as well as electric multiple unit trains and electric push pull trains. On their New Haven Line the trains switch between third rail and overhead wire power. East of Hudson service is run by Metro North on 3 lines. There are also three abandoned lines:
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and West of Hudson service is contracted to New Jersey Transit to operate
New Jersey Transit Runs electric multiple unit, electric push pull and diesel push pull. They serve New Jersey and New York. One line actually starts in Philadelphia before returning to New Jersey.
Shore Line East runs from New Haven to New London in Connecticut and once ran to Boston, Massachusetts
New York Westchester and Boston (NYWB) is now abandoned and lives on in portions of the New Haven Line and East 180th Street on the 2 and 5 subway lines. See New York Westchester and Boston White Plains and New York Westchester and Boston Port Chester Pages.
* denotes abandoned line