The Port Authority Trans Hudson (better known as PATH), is odd as far as subways in the United States go. Operated by the Port Authority of New York and New Jersey (who you may better known for operating the crossings over the Hudson River or the New York metropolitan area's much-maligned airports), the PATH is actually descended from a railroad known as the Hudson and Manhattan. As such, it has more in common with suburban rail, as the system runs between New York City proper and several cities in New Jersey. Nonetheless, it's still a key system in a metropolitan area where real estate is real hard to come by. If you can finagle a transit plan with it, it'll save you money too, since single-rides are $2.75 - the toll on every single bridge and tunnel across the Hudson River is $15note . It also runs 24/7, albeit on a very infrequent schedule late at night.
It has two main terminals on each side of the Hudson - in New York they're the World Trade Center in Lower Manhattan and 33rd Street (right next to New York Penn Station) in Midtown, while on the Jersey side they're Hoboken and Newark Penn Station. Yes, there are two major stations in the area called Penn Station, distinguished by cities whose names are pretty easily mistaken for one another.note During normal weekday rush hour, four lines are run that generally begin and end at each of these four terminals (the exception is the 33rd Street-Newark line, which only goes as far as Journal Square on the near side of Newark Bay because Newark and Harrison are significantly more distant than the other stations and the train makes multiple stops in Manhattan - doing both would make trips too long). Overnight and on weekends, two runs are maintained (Newark-WTC and Journal Square-33rd St. via Hoboken).
It has a close cousin in the Port Authority Transit Corporation (PATCO) Speedline, a subway run by the Philadelphia equivalent of the Port Authority, the Delaware River Port Authority, which connects Center City Philadelphia to its New Jersey suburbs (from Camden to Collingswood and Haddonfield), and similarly runs 24/7.