"The 9mm" (as in the typical 9mm) is the 9mm Luger. As noted it's known by many names.
9X19mm (diameter and length of cartridge), 9mm Luger (after the inventor of both the round and the pistol) 9mm Auto. 9mm Parabellum (the other common name) or "9 Para".
In Germany for military use it used to be called the Patrone '08 (cartridge used in the P08, the Luger). All the same ammunition.
Various other 9mm cartridges which will NOT ( I REPEAT NOT ) work in a Glock 19:
9X17mm (9mm Kurz, 9mm Browning Short, .380 ACP or Automatic Colt Pistol) which is shorter.
9X18mm Makarov (also shorter just slightly) primarily used in the Makarov Russian service pistol and a few others.
9X23mm Bergmann-Bayard (the old Spanish service round)
9X20mm (I think) Browning Long (you'll highly unlikely EVER find this round)
9X21mm IMI (There are Glock 19s chambered in this round, but only for European usage, for countries where military rounds are not for civilian usage)
9X25mm Dillon (competition round)
Most of those rounds you'll never find except in odd corners of a large gunshop. Typically just make sure it's 9mm Parabellum/Auto/Luger and you're good.
For range use, shoot FMJ (Full metal jacket, like the military uses) it's cheap and does fine. For home defense/personal defense use hollowpoints (JHP, jacketed hollowpoint) like the Federal HST, Hydrashok, Remington Goldensaber, Winchester SXT, etc.
Do not use plain lead (no jacket at all) bullets in a Glock. Since most shops sell FMJ for range use, not a real problem, but just so you know. The rifling in Glocks and plain lead rounds do not agree and will shave some off, eventually building up to where you can overpressure and make your gun go kaboom. Not a problem with any jacketed ammo.
9X19mm (diameter and length of cartridge), 9mm Luger (after the inventor of both the round and the pistol) 9mm Auto. 9mm Parabellum (the other common name) or "9 Para".
In Germany for military use it used to be called the Patrone '08 (cartridge used in the P08, the Luger). All the same ammunition.
Various other 9mm cartridges which will NOT ( I REPEAT NOT ) work in a Glock 19:
9X17mm (9mm Kurz, 9mm Browning Short, .380 ACP or Automatic Colt Pistol) which is shorter.
9X18mm Makarov (also shorter just slightly) primarily used in the Makarov Russian service pistol and a few others.
9X23mm Bergmann-Bayard (the old Spanish service round)
9X20mm (I think) Browning Long (you'll highly unlikely EVER find this round)
9X21mm IMI (There are Glock 19s chambered in this round, but only for European usage, for countries where military rounds are not for civilian usage)
9X25mm Dillon (competition round)
Most of those rounds you'll never find except in odd corners of a large gunshop. Typically just make sure it's 9mm Parabellum/Auto/Luger and you're good.
For range use, shoot FMJ (Full metal jacket, like the military uses) it's cheap and does fine. For home defense/personal defense use hollowpoints (JHP, jacketed hollowpoint) like the Federal HST, Hydrashok, Remington Goldensaber, Winchester SXT, etc.
Do not use plain lead (no jacket at all) bullets in a Glock. Since most shops sell FMJ for range use, not a real problem, but just so you know. The rifling in Glocks and plain lead rounds do not agree and will shave some off, eventually building up to where you can overpressure and make your gun go kaboom. Not a problem with any jacketed ammo.