What is the best sub machine gun in the world?

Sub machine guns usually fire pistol ammunition, as opposed to assault rifles, which fire rifle ammunition. For this reason sometimes submachine guns are called machine pistols. Rifle cartridges are longer, to contain more propellant and are capable of generating higher muzzle velocity.

Probably the largest caliber submachine gun ever manufactured was the Thompson submachine gun, which was also one of the first. It entered production just barely too late to see service in WWI. General Thompson, its designer, intended it as a "trench broom". The Thompson was a .45 caliber weapon, firing the same .45 ACP (Automatic Colt Pistol) ammunition as the Model 1911 .45 pistol, used by the US Military into the 1980s.

During WWII the US manufactured the M2 submachine gun as well. This was a much cheaper weapon to manufacture than the Thompson, a very simple design, which also fired the .45 ACP. Its often called the "grease gun".

The US also developed one of the first assault rifles for WWI, which again was in production a little too late for much service at the front. This was the Browning Automatic Rifle (BAR), which remained in service through the end of the Korean War. It fired the same .30-06 ammunition as the standard rifles of the US military up through the M-14. This was also the same cartridge used in belt-fed US .30 caliber machine guns.

added:

While the above information would be correct about the .45 ACP being the largest calibre cartridge to see at least any noteworthy usage in a submachinegun, a lot of the information posted above is highly inaccurate.

• The Thompson submachinegun was not the first submachinegun designed. The first true submachinegun was the Bergman MP18, which predated the Thompson by some three years. Before either of those designs, a number of full auto variants of pistols had been manufactured and marketed, albeit unsuccessfully.
• The definition of a submachinegun varies between sources, including manufacturers. Heckler & Koch marketed their HK53 and LaFrance marketed their M16K as submachineguns, just as the AKS-74U is considered to be a submachinegun to some sources, and assault carbines by other sources. All three of them (and several other weapons sometimes referred to as submachineguns) fire full power rifle cartridges.
• The Browning M1918 BAR was not an assault rifle - it was designed from the onset to fulfill the role of a light machine gun, in response to the French 'walking fire' tactic, and it was not the first weapon in its class - the Soviet Union fielded the rifle designed by Colonel Federov from 1916 tp 1925, which was much closer to the assault rifle concept - it featured selective fire capabilities, was designed to be used as an individual weapon, and used a reduced power cartridge (the Japanese 6.5 Arisaka). However, the first true assault rifles did not appear until well into the Second World War with the StG-44/MP-43. This became the basis of future assault rifle designs - an intermediate cartridge (as opposed to a full power rifle cartridge), fire from a weapon with select fire capability, and intended for engagements of 300 metres or less.
• The nomenclature for the "Grease Gun" was M3, not M2.