A spitzer bullet is a bullet with a pointed, aerodynamic tip. The word 'spitzer' comes from the German word spitzgeschoss, which in typically literal Germanic manner translates roughly to 'pointed bullet' (spitz: adj, 'pointed' or 'sharp' and geschoss: noun, 'projectile' or 'piece of ammunition fired into the air.')1
Bullets were originally spherical, the 'ball' of Ball ammunition, since they were loaded separately and since originally their orientation didn't matter. It wasn't until the advent of rifled barrels that projectiles began to become shaped especially for the job. The famous Minié ball, invented in 1848, wasn't a ball at all but was an expanding projectile of a shape quite familiar to modern shooters, with a conoidal front.
Even then, the very tip of the bullet was usually shallowly rounded as the sphere ancestry made itself known, or even cut off flat for either manufacturing or damage purposes. Even those which were pointed had very wide fronts, as it was easier to deform a sphere into that shape and the round was less fragile. Prior to jacketed ammunition, bullets were made entirely of soft lead, which wouldn't hold a point well anyway, and might bend if formed into a sharp tip which would make the bullet unstable.
Once jacketed bullets became the norm, however, it was clear that sharply tipped bullets could be manufactured and used easily. In 1886, the French army adopted the first smokeless powder cartridge design to be used by any country, the 8x50 mm Lebel cartridge, named for Lt. Col. Nicolas Lebel who invented the flat-tipped wadcutter bullet it used, known as the Balle M. The main reason for the flat tip was to ensure good performance in the rifle's tube magazine, making sure the bullets remained in a straight line and didn't deform at the front where they met the cartridge ahead of them.
In 1898, the Balle M was superseded by a new design (the 'Balle D') which had a pointed tip and a boat-tail. This design was proven to have much better aerodynamic performance, hence increased range and accuracy. At the time, armaments advances were closely studied by neighboring countries, and such advances could hardly be kept secret. The German army, upon testing the Lebel cartridge with Balle D, decided the advantage was noteworthy and modified their standard 8x57J cartridge to include a pointed bullet, naming the result the 8x57JS - 8mm across, 57mm long, Jaeger, Spitzgeschoss (Jaeger means, roughly, rifleman or 'rifle infantry'). This round would become known as the '8mm Mauser' and Germany used it through both World War 1 and World War 2.
A few years later, the U.S. Army licensed the spitzgeschoss bullet design from its German designer, Herr Arthur Gleinich - and U.S. personnel quickly shortened the unfamiliar and difficult German word spitzgeschoss into the americanized adjective spitzer.
1: Both from Google translate.