"Luger", originally, meant the Luger P08 pistol. You'd probably recognize it as the sidearm you see the Nazis carrying in WWII movies. It was the first pistol to take the cartridge which today most people just call "9mm".

Today, "9mm Luger", "9mm Parabellum", and "9x19mm" are all synonyms for this same cartridge. Usually when "9mm" is said without qualification, this is what it means. However, there are other much rarer 9mm cartridges such as the 9x18mm Makarov, and also "9mm short" and "9x17mm" are rarely-used synonyms for the (common) cartridge usually called .380ACP.

No, but the Luger, or 9x19mm Parabellum as it's also called, is by far the most common type of 9mm ammunition used.