one of the alternative names for 9x19mm Parabellum used by numerous pistols, sub machine guns and pistol-calibre carbines. the origional was developed by the germans for the 9mm luger pistol, hence the americans also knowing the round as 9x19mm luger. it's all the same thing.
The NATO part comes from the fact that it is the standard pistol bullet used by the armies of NATO countries, such as the USA, Britain, Canada and France (to name but a few)
notable guns which use this round include the classic WW1/2 German Luger, The H&K MP5 series, The UZI, The Browning Hi-Power (P38) and some glock models.
person 1: hey man, is that glock using .45 ACP?
person 2: nope, it uses 9x19mm NATO