9x17 is the metric designation for the cartridge generally known in the USA as .380 ACP (for
Automatic
Colt
Pistol) or .380 Auto. Developed by the famous firearms designer John Moses Browning, it's commonly known as 9mm Browning Short in Europe, with "Short" often being translated into the local language, e.g. Court in French, Kurz (pronounced "Curts") in German, or Corto in Spanish and Italian.
The reason it's "Short" is to differentiate it from a previous and longer Browning cartridge, 9x23mmSR aka 9mm Browning or .38 ACP. This cartridge was used in the Colt Model 1900 and has largely fallen into disuse, although it's the parent cartridge to .38 Super.
9x18 refers to one of two dissimilar cartridges. The most common is 9mm Makarov, a cartridge developed by the Soviets shortly after WWII for use in their new PM (for "Pistolet Makarov") service pistol. This cartridge was also used in several other Warsaw Pact pistols common on the surplus market. It's popular enough to several companies currently market the ammo.
The other 9x18 cartridge is the 9mm Ultra and the closely related 9mm Police. It was developed by the Luftwaffe prior to WWII but was never adopted because 9x19mm Parabellum and 7.65mm Browning (.32 ACP) were deemed adequate for all Nazi pistol needs. (In the 1930s, the Luftwaffe conducted several unnecessary programs to develop redundant bespoke equipment, largely due to the exaggerated political influence of Hermann Göring, but I digress.
). The cartridge was resurrected in the late 1960s in an effort to field a German national police cartridge more powerful than 9x17/.380 but easier to shoot than 9x19, but the authorities ultimately (and rightly IMHO) decided that simply adopting 9x19 was a better idea since off-the-shelf pistols and ammo were readily available, and most people can shoot 9x19 reasonably well with appropriate training. Most of the surplus 9x18 Police pistols and ammo were offloaded on the U.S. commercial market. (Today it's much easier to find the pistols than the ammo, since the pistols were never common enough to make it worthwhile for the ammo companies to produce it.)