I am mostly into shotguns and archery but just love the new Springfield Armory XD. I buying one for my wife (sub compact 3") and was wondering about the 9mm cartridge.

When people speak of the 9mm luger/parabellum they are referring to the same thing. Not sure where luger came from other than the name of the pistol itself. The 9x19 no idea where that came from either.
The 9mm is a good cartridge for plinking/self defense. It is not a deer hunting cartridge although could kill a deer with proper placement. The 9mm has killed several people throughout history and continues to be in use by our military. The new designs in powders and hollowpoint bullet technology makes it a good choice for someone who wants high capacity and low recoil. I would use the 124 grain bullet or 147 grain in Speer Goldots, Remington Golden Sabers, Winchester Ranger SXT's, or Federal HST's. Any of these would be ideal for self defense. Esox357.

The 9mm is the same as the 9x19, 9mm luger and parabellum.

The 9x18 is the Makarov round, so def don't get those confused.

Lastly, the 9x17 is the .380, AKA 9mm short or, what it was originally called in Germany, the 9mm kurz.

9x19 cartridge for the Parabellum (pistol for war) 1908 (P-08) designed by Geo Luger.

The rounds in the post title are identical and interchangeable.

There are a host of other 9mm rounds that are NOT the same and may NOT be interchanged. Just a partial list of sometimes-encountered rounds include the 9mm Bergman, 9mm Largo, 9mm Makarov, 9mm Ultra, 9mm Federal, 9mm Win Mag - and many more!

The safest approach is to buy and use ONLY ammunition that exactly matches the gun's rollmark designation.

Let us not forget the 9x21 and the 9x23 which also are certainly not interchangeable with the 9x19.