Looking at a cartridge from a few feet away, anyone who's a gun geek and has decent eyesight can probably tell you that a 9mm cartridge is either that, a .40 S&W, or something very obscure. Also though, almost all handgun cartridges have their caliber stamped on the rim. I just checked to see from how far away I can read the markings on my Federal .40 (I don't own any 9mm, but it should be basically comparable). In good light and at just the correct angle, I can read it from about arm's length. Otherwise I have to hold it close.
"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.
