The 5.56x45 mm cartridge is a NATO standard, and extremely common around the world - in the US it is also known as the .223 Armalite/Remigton.

The 5.56 (.223) was developed from the larger and heavier 7.62x51 mm NATO ( .308 Winchester ), in the effort to produce a weapon/ammunition combination that would allow the soldier a fast-firing rifle for close quarter combat, as well retaining accuracy at medium distance combat since battlefield investigations had shown that most combat actually takes place within 300-400 meters.

The ammunition developed had at first a light bullet at 55 grains, which resulted in a high muzzle velocity at 3250 feet/second, but also a fairly short effective range. Later loadouts would deploy a heavier bullet (69 grain) to gain effectiveness at a greater range - but now sacrificing muzzle velocity, now at 3000 feet/second.

The 5.56 is now the favored military ammunition, with its low weight and size - resulting in a smaller, more manageable weapon for the soldier.
5.56 is also the caliber of choice among law enforcement snipers, many agencies consider the very popular 7.62x51 mm round being too powerful, and could over-penetrate and injure innocent bystanders/hostages. The 5.56 bullet tends to break up on impact; allowing almost no over-penetration. This breakup obviously calls for a better shot placement, and as the 5.56 bullet has a small temporary wound channel it requires almost a direct hit on the spinal stem to bring down a target instantaneously.
Although the 5.56 has a very flat trajectory, which gives it great accuracy it proves close to unusuable as a military sniping round, since the light bullet is very sensitive to wind. For military purposes, an incapacitating shot is hard to achieve due to lack of bullet energy and penetration.

For a sportsman's weapon, the 5.56 is probably one of the best calibers available. It has almost no recoil, is easy to shoot nice series, and fun. The 5.56 is an excellent varmint round and compared to the .22, it offers hunting opportunities for small game that the .22 cannot handle in terms of accuracy and killing power.
Larger game such as large boar and deer can be attempted if the shooter is skilled enough, but is generally a bad idea. These targets should be left to the larger, 7.62 mm caliber series, ammunition.