First, let's talk about calibers. There are two general ways to state calibers - standard and metric. A 7.62mm is a .30 caliber round. The .30-06, the .308, the 300 Winchester Magnum, the 7.62x39 - these are all firing the same bullet. Same thing with the .357/38 and the 9mm, the .40 caliber and the 10mm, and the .45/12mm. Now, this only refers to the bullet, not the cartridge.

Cartridges are the brass case that holds the primer, the powder, and the bullet in one convenient package. For example, the 7.62/.30 caliber was the most common bullet used by militaries until the 60's and 70's. The United States used the M1 Garand chambered in 7.62 x 63 (or .30-06 Springfield), and the M14, chambered in the 7.62 x 54 NATO (.308 Winchester). The Communists used the 7.62 x 39 (known as the 7.62 Soviet) in the AK-47. In the metric measurements, the second number is the length of the case in mm. The .30-06 is the longest and can hold the most powder and the Soviet round is the shortest. The cases were designed to hold different amounts of powders, to chamber into different guns, and to be used in different manners. The .30-06 is an effective round out to 1,000 yards, while the Soviet round is only useful for about half that distance. These days, military doctrine has shifted so that most modern militaries are using a .22 caliber round. The US and NATO use the 5.56x45/.223 Remington, while the Russians use the 5.45x39 round and the Chinese have the 5.8x42 round.

As to the common convergence around certain calibers, it all comes down to ballistics.

In the late 1800's, the US military used a .38 for their sidearm. During the Spanish-American War, US soldiers told stories of unloading their entire gun into Filipino soldiers who were charging them. The Filipinos would die, but only after reaching the US soldier and hacking him to death with their swords! The .38 was not effective against the Filipino guerrillas who were getting high before the battle - think about the stories of PCP users attacking the police and you get the idea. So the US military began to seriously investigate ballistics. The Thompson-LaGarde tests involved shooting cattle and dead bodies at the Union Stockyards in Chicago. The .45 was determined to be the optimum round and was subsequently adopted by the US military in the form of the M1911 .45 Automatic made by John Browning.

Rounds have many different properties that have to be considered. The two main ones are penetration and expansion. If you have a round that penetrates very well, it might easily go out the back of your target and into something else. This is a concern for police units when they are engaged in firefights inside of homes - drywall does not stop a bullet very well and if a round goes through a person, it could easily penetrate a wall and hit another person. Expansion is the ability of the bullet to deposit energy. A bullet that expands rapidly will deposit energy very efficiently.

These two properties are directly related. Any energy used to penetrate does not cause expansion and vice-versa. The Thompson-LaGarde tests determined that a .45 caliber round was the best compromise between these two characteristics. Smaller, faster bullets (9 mm) have better penetration while larger, slower bullets (.45) have better energy deposit. Please note, this is an overly simplistic explanation. Other characteristics - fragmentation of the round, hydraulic shock, wound channels - all play a part in this as well, but they are irrelevant to the question at hand.

So, why do some common calibers keep getting used? Battle doctrine. The US military switched to the M16 during the Vietnam War. The M16 weighs 3.26kg unloaded, 3.99 kg loaded. The M14 weighs 4.1kg unloaded, 5.2kg loaded. The M1 Garand weighs a bit more than the M14. The difference here is that the M16 and M14 were both select fire while the M1 was semi-auto only. A .30-06 and a .308 both have a bit of a kick to them and attempting to fire these rounds in an automatic rifle is incredibly tough. The recoil causes the rifle to climb and you quickly lose sight of your target. The .223/5.56 round has a much lighter recoil, allowing for better control during automatic fire.

Further, you can see that the weight difference is substantial for a loaded rifle. The .223/5.56 round is much lighter than a .30 caliber round. It carries less energy than a .30 caliber round, but it travels at a higher velocity, negating some of this loss of energy. The modern battle doctrine is one of lightweight weapons and rounds so that soldiers can carry more ammunition.

There are some complaints and other issues with the 5.56. Cartridges are designed to work optimally with a certain configuration. For the 5.56, it's designed to achieve maximum output in a 20 inch barrel. The US military has been getting away from the M16 and moving to the M4 - same gun, shorter barrel. The problem is that the 5.56 is a great round when it's being used optimally, but in a shorter barrel configuration it loses a lot of power and velocity resulting in less terminal shots, i.e. people aren't incapacitated immediately. This is partly why, recently, we have seen the development of newer rounds like the .458 SOCOM, a powerful cartridge to be fired from the M16 platform with incredibly short barrels and a suppressor. But I digress...

Prior to the lightweight doctrine, the idea was that battles with rifles would take place at distance and the .30 caliber round was suited for that. The US made .30-06 and the German 8mm Mauser both fit that bill, being suitable for upwards of half mile long shots.

So, we see the same bullet sizes because they perform in a fairly standard way out of similar cartridges. The way that they perform is useful for different battle doctrines that have been adopted by militaries around the world. The early 20th century doctrine was one of power and distance (.30 calibers), the later 20th century was one of light-weight, controllable automatic weapons and lots of ammunition. (.22 calibers) After Somalia and other US experiences, there has been a push for heavy, powerful rounds that can be suppressed for use in urban combat. (.458 SOCOM)

I think we may have missed the primary point of the OP. No question that the responders are right, as regards multinational use/alliance use/etc., and as regards changing tactical doctrines which have led to a reduction in calibers over time (.30 giving way to .223, etc.) and wide adoption of standard cartridges. However, I think a more interesting question is why .308 bullet diameter became so much more popular than, for example, .313, or .309, or .306. There are lots of possible bullet diameters, but only a few seem to have been adopted widely. I would be interested in finding out if there is a perceived "sweet spot" for bullet diameter which provides improved accuracy (or reliability, etc.) with diminishing benefits if the diameter is changed, either up or down. For example, in handguns, .32 is common, but there are very few bullets of diameter .31, .33, or .34. In rifles, there are lots of .223 bullets, but why are there so few .227 bullets in common use? In the early years of cartridge development there were many bullet diameters used, but only some have persisted-- Is this all due to the adoption of such a standard by a major military, or are there ballistic reasons why an "oddball" diameter simply doesn't work as well? I am aware that the British widely used the .303 (which is really a .312), and the Japanese in WW2 used the 7.7 mm (which is also a .312), but why is there not a common use of a (for example) .313 or .314?

It mostly boils down to international alliances, such as NATO. The reason the "5.56 NATO" round is called such is inferable by its name - in joint, multinational operations, the inability to share munitions would be a logistical nightmare. Regardless of the type of gun being used or the country of origin, the same bullet can be fired, making international missions between NATO countries and other respective alliances much easier.