The .38 Special is a straight-walled, rimmed revolver round. It was originally created by S&W by lengthening the .38 Long Colt case and loading to higher velocity. The first firearm to chamber this cartridge was the S&W Military & Police K-Frame Double Action revolver (this later became the Model 10 when S&W went to model numbers in the 1950's) in 1902. Originally designed for both smokeless and blackpowder loadings, the .38 Special has a very large case capacity (much more than is needed for the velocities attained with smokeless powder). While there have been a few semi-automatics chambered for the cartridge, it is far more preavalent in revolvers. Nearly every major revolver manufacturer has chambered a revolver in this caliber at one time or another and the the cartridge is very popular being a favorite among both police and civillians for decades. Bullet diameter for this cartridge ranges from .357-.359 inches and typical bullet weights range from 110grn to 158grn although lighter or heavier bullets may be encountered. The traditional loading for this caliber is a 158grn lead round nose bullet at 750-800fps from a 4" revolver barrel although a wide variety of bullet types, bullet weights, and velocities are widely available.
The .380 ACP (also known as 9mm Short, 9x17 Browning, 9mm Kurz, or 9mm Corto) is a straight-walled, rimless semi-automatic cartridge. The cartridge was introduced by Colt and first-chambered in the John Browning-designed Colt M1908 Pocket Automatic (not to be confused with the Colt M1908 Vest Pocket Automatic in .25 ACP). While there have been a few revolvers that could chamber this round (primarily those such as the Medusa which could fire a number of .38 caliber rounds), this cartridge is primarily found in small semi-automatics. Bullet diamter for this cartridge is .355 inches and typical bullet weight ranges from 88grn to 102grn although both lighter and heavier bullets may be encountered. The traditional loading in this caliber is a 95grn Full Metal Jacket round nose bullet at 950-1000fps from a 4" semi-automatic barrel though like the .38 Special there is a wide variety of ammunition available. This cartridge is popular in the United States in small concealment or backup type guns while in Europe is was popular for decades as both a Police and even Military handgun, though in the latter role it has been overshadowed in the last few decades by the 9x19 Parabellum cartridge. The .380 ACP is typically considered to be one of the most powerful cartridges suitable for use in an unlocked blowback semi-automatic (an action type most commonly encountered in small, concealment type guns), but in more recent years there have been a number of locked-breech .380's such as the Kel-Tec P3AT, Ruger LCP, and Sig P238 because a locked-breech allows a smaller and lighter gun than a blowback such as a Walther PPK, Sig P230, or Bersa Thunder can.
A note about bullet diamter, while both .38 Special and .380 ACP are approximately 35 caliber, they and many other 35 caliber rounds like 9x19 Parabellum, .38 Super Auto, and .357 Magnum are typically referred to as 38 Caliber. The reason of this has to do with measurement of heel-based and inside lubricated bullets. Metallic cartridges were originally loaded with heel-based bullets whose bearing surface was the same diameter as the outside of the case and had a smaller-diamter heel that fit inside the cartridge case. When the move was made to inside-lubricated bullets (same diameter from base to ogive) the cartridge case remained constant while the bearing surface of the bullet was reduced. Even though bullet diameter was reduced, the names of the cartridges were not. Hence a 44 caliber is actually .429" in diameter, a 32 caliber is actually .311, and a 38 caliber is actually .355 to .358.
With regards to +P ammunition, this is a designation recognized by SAAMI (then entity that governs manufacturing standards for most U.S. made ammunition) in .38 Special, 9x19 Parabellum, and .45 ACP. Officially, a +P designation means that a cartridge can be loaded up to 10% over its maximum standard pressure. +P+ ammunition and +P cartridges other than .38 Spl, 9mm, and .45 ACP are not officially recognized by SAAMI and thusly may be loaded any way the ammo-maker sees fit. CIP (the European counterpart to SAAMI) does not recognize +P or +P+ ammunition in any caliber though their maximum standard pressures are slightly higher in several calibers.