Smoothbore means that the barrel is smooth, not rifled. This is used to achieve very high projectile velocities while avoiding barrel wear. Projectile stabilization is achieved either with fins, or with spiral grooves cut into projectile itself that spin it by releasing a portion of gases forward (to my knowledge, this system was never operationally used, but I know for a fact that it was developed and worked - my father was part of the team that made it during WW2).
Most common military ammunition... well, on small arms, here goes.
NATO:
9mm Para - also known as 9x19mm Luger/Parabellum. Based on 7.62x25mm round (.30 Mauser), originally developed for German Navy who weren't satisfied with a 7.62mm round in a handgun. Currently, it is standart NATO cartridge for handguns and submachineguns.
5.56x45mm NATO, also known as .223 Remington. Developed for the AR-15 (M16) rifle, currently it is standart NATO assault rifle cartridge. Original cartridge had a lead core copper jacketed FMJ bullet, US military designation M196. In early 80's, a new bullet was adopted, with a steel penetrator in nose, requiring a barrel with a heavier twist (1 turn in 7 inches, instead of 1 turn in 12 inches for the original), under US military designation M855. Despite the small size, the cartridge is extremely lethal, due to bullet being very unstable and tumbling when entering flesh. M196 cartridges are unmarked, M855 cartridges are marked with a green tip.
7.62x51mm NATO, also known as .308 Winchester. Based on the .300 Savage hunting cartridge, it replaced .30-06 in US military in 1953, but turned out to be too strong for controlled automatic fire in battle rifles (M14, FN FAL) and is now used mostly in sniper rifles and medium/general purpose machine guns.
12.7x99mm, also known as .50 BMG (Browing Machine Gun). Developed in early 1920's for the Browning M2 heavy machine gun which is still operational with many military forces around the world, it is also used in heavy anti-material sniper rifles.
Eastern Block:
9x18mm Makarov. Based on German 9x18mm Ultra round (though not interchangeable, as Makarov bullet is 0.2mm larger. Adopted, together with the PM (Pistolet Makarova) pistol as official Red Army sidearm in 1951. Since then, used in many other weapons, modernized several times.
7.62x39mm Intermediate. Developed in 1943, first used in SKS (Samozaryadnyi Karabon Simonova - Simonov Self-loading Carbine), then in AK (Avtomat Kalashnikova) and RPK (Ruchnoi Pulemet Kalashnikova) series. Officially, this cartridge has been replaced by 5.45x39mm, but in fact, it remains widely used.
5.45x39mm. Adopted in 1974, together with AK-74 and RPK-74 series. Basically, an attempt to follow Americans with their 5.56x45mm cartridge. Original bullets were very ineffective, leaving troops very dissatisfied with the caliber both in Afghanistan and Chechnya, though newer bullets with improved penetration are supposed to have solved that problem.
7.62x54mm Rifle. Originally developed in late 19th century for the Model 1891 Mosin rifle, modernized in 1908 with a new sharp-nosed bullet. Still in use by SVD (Snayperskaya Vintovka Dragunova - Dragunov Sniper Rifle), PK/PKT machine guns, GShG aircraft machine gun, and others.
12.7x109mm... sorry, but I know very little about this round. To my knowledge, it was originally developed for the DShK heavy machine gun in early 1930's, later used by NSV and Kord heavy machine guns, as well as some heavy sniper rifles.
14.5x114mm. Developed in early 1940's for anti-tank rifles, used by PTRD (Protivotankovoye Ruzhiyo Degtiariova - Degtiariov Anti-Tank Rifle) and PTRS rifles, later used by KPV/KPVT super-heavy machine gun, which is most often used as the primary weapon of BTR-70, BTR-80, and BRDM-2.
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Sorry, this isn't very informative, but it's the best I could come up with on a short notice.